# Explicit Solution of the Inverse Eigenvalue Problem of   Real Symmetric Matrices and Its Application to Electrical Network Synthesis

Authors: D. B. Kandić; B. D. Reljin
Journal: Mathematical Problems in Engineering
 (2008)

---

## Abstract

A novel procedure for explicit construction of the entries of real symmetric matrices with assigned spectrum and the entries of the corresponding orthogonal modal matrices is presented. The inverse eigenvalue problem of symmetric matrices with some specific sign patterns (including hyperdominant one) is explicitly solved too. It has been shown to arise thereof a possibility of straightforward solving the inverse eigenvalue problem of symmetric hyperdominant matrices with assigned nonnegative spectrum. The results obtained are applied thereafter in synthesis of driving-point immittance functions of transformerless, common-ground, two-element-kindRLC networks and in generation of their equivalent realizations.


---

## Body

## 1. Introduction

During the past few decades, many papers [1–16] have
studied the inverse eigenvalue problems (IEPs) of various types. The solution
existence of the specific IEPs was generally considered in [1, 3–8, 10, 11, 13, 14]
without explicit formulation of the corresponding procedure for solution construction,
whereas in [2, 9, 12, 15, 16] this has been accomplished. The main result of
[16] is the proof that IEP of symmetric hyperdominant (hd) matrices with
assigned nonnegative spectrum has at least one solution which has also been constructed.
This settled an old IEP opened in [17]. Hyperdominant matrices have nonnegative
diagonal and nonpositive off-diagonal
entries and nonnegative hd margins of rows (hd margin of a row is the sum of
entries in that row). The tool used in [16] to construct the
nth-order
hd matrix with assigned spectrum was the nth-order
orthogonal Hessenberg matrix constructed as a special product of n
−
1 plane rotations [15]. Hessenberg matrices naturally arise in study of symmetric tridiagonal matrices, skew symmetric, and
orthogonal matrices [13, 14, 18]. A matrix is upper (lower) Hessenberg if its entry
(k, m) vanishes whenever k
>
m
+
1
(
m
>
k
+
1
).In practical
work, it is commonly assumed to be better not to form Hessenberg matrices explicitly,
but to keep them as products of plane rotations. On the other hand, explicit
construction of real symmetric matrices with nonnegative spectrum, which either
have hd sign pattern or are truly hd, is proved to be an inevitable task in
considering the synthesis of driving-point immittance functions of passive,
transformerless, common-ground, two element-kindR
L
C networks and in generation of their equivalent realizations
[17–19]. R
L
C networks are comprised
solely of resistors (R), inductors (L), and capacitors (C). Driving-point immittance function of a lumped, time invariant,
linear electrical network is either a driving-point impedance Z
(
s
),
or a driving-point admittance Y
(
s
)
=
Z
−
1
(
s
) (s
=
σ
+
j
⋅
ω is the complex frequency; σ, ω are real numbers; j
:=
−
1). It is well known that a real rational function in s can be driving-point immittance function
of R
L
C network if and only if it is positive
real function in s; or similarly, a necessary condition for a stable square matrix W
(
s
) of real rational
functions in s to be driving-point immittance
matrix of a passive R
L
C network is
that W
(
s
) be positive real matrix [20, 21]. A few tests for ascertaining positive
real properties of functions and/or
matrices can be found in [20, 21]. In [22] it has been pointed out the role of hd
matrices in synthesis of both passive and active, transformerless, common-ground
multiports. Unlike [16], this paper presents explicit construction of entries of
real symmetric matrices with arbitrarily assigned spectrum and the entries of the
corresponding orthogonal modal matrices. It also presents explicit construction
of real symmetric matrices with assigned spectrum and with specific sign
patterns (including hd one). Thereof, a solution to the IEP of symmetric, truly
hd matrices with assigned nonnegative spectrum is produced. Some of the obtained
results are then applied in synthesis of driving-point immitances of
transformerless, common-ground, two-element-kind R
L
C networks and in generation of their equivalent realizations. The
two proposed realization procedures are illustrated by an example. Throughout
the paper ⊕ denotes direct sum, x
T denotes transpose of x, bold capital letters denote matrices and I
k stands for
the kth-order unit or identity
matrix.


## 2. Explicit Solution to the IEP of Real Symmetric Matrices by Using Canonic Orthogonal Transformations

Let{
λ
1
,
λ
2
,
…
,
λ
n
} be assigned
spectrum of the sought real symmetric matrices and let G
1
:
=
diag
(
λ
1
,
λ
2
,
…
,
λ
n
) be n
×
n spectral
matrix. Consider a set of 2
×
2 orthogonal
matrices P
k
∈
{
A
k
,
B
k
,
C
k
,
D
k
}
(
k
=
1
,
…
,
n
−
1
):
(2.1)
P
k
:
=
[
a
k
b
k
c
k
d
k
]
,

A
k
:
=
[
cos
θ
k
−
sin
θ
k
sin
θ
k
cos
θ
k
]
,

B
k
:
=
[
cos
θ
k
sin
θ
k
sin
θ
k
−
cos
θ
k
]
,
C
k
:
=
[
cos
θ
k
sin
θ
k
−
sin
θ
k
cos
θ
k
]
,

D
k
:
=
[
−
cos
θ
k
sin
θ
k
sin
θ
k
cos
θ
k
]
,

θ
k
∈
[
0
,
π
2
]
,which are
either rotators (A
k and C
k) or
reflectors (B
k and D
k). A useful set of n
×
n orthogonal matrices is
(2.2)
U
1
:
=
P
1
⊕
I
n
−
2
,


U
k
:
=
I
k
−
1
⊕
P
k
⊕
I
n
−
k
−
1

(
k
=
2
,
…
,
n
−
2
)
,


U
n
−
1
:
=
I
n
−
2
⊕
P
n
−
1
.
From the following two matrix recurrent relations
(2.3)
G
k
+
1
:
=
U
k
G
k
U
k
T
,

S
k
+
1
:
=
U
n
−
k
S
k
U
n
−
k
T

(
k
=
1
,
…
,
n
−
1
),
we readily obtain n
×
n real symmetric matrices G
n and S
n, which are both congruent and similar to G
1
(2.4)
G
n
=
U
G
1
U
T
,

U
:
=
U
n
−
1
U
n
−
2
⋯
U
1
;


S
n
=
V
G
1
V
T
,

V
:
=
V
1
V
2
⋯
V
n
−
1
.
Columns of the orthogonal modal matrix U
(
V
) correspond to eigenvectors of G
n
(
S
n
). Out of (
n
−
1
)
!
different possibilities of using (2.3) in generation of G
n and S
n,
only the two selected by (2.4) produce explicit expressions of entries of G
n and S
n in terms of {
λ
1
,
λ
2
,
…
,
λ
n
} and the entries
of P
k
(
k
=
1
,
…
,
n
−
1
). U
(
V
) from (2.4) will be
shown later to take on lower (upper) Hessenberg form with the entries explicitly
expressed too. For the sake of brevity, we will restrict our consideration only
to the first of relations (2.3), bearing on mind the possibility of treating the second
one similarly. For k
=
1 and k
=
2 we readily obtain G
2 and G
3, by using (2.1) and (2.3):
(2.5)
G
2
=
(
P
1
⊕
I
n
−
2
)
G
1
(
P
1
T
⊕
I
n
−
2
)
=
[
a
1
b
1
c
1
d
1
0
2
,
n
−
2
0
2
,
n
−
2
T
I
n
−
2
]
[
λ
1
0
0
λ
2
0
2
,
n
−
2
0
2
,
n
−
2
T
λ
3
0
⋱
0
λ
n
]
[
a
1
c
1
b
1
d
1
0
2
,
n
−
2
0
2
,
n
−
2
T
I
n
−
2
]
=
[
λ
1
a
1
2
+
λ
2
b
1
2
(
λ
1
−
λ
2
)
a
1
c
1
(
λ
1
−
λ
2
)
a
1
c
1
λ
1
c
1
2
+
λ
2
d
1
2
0
2
,
n
−
2
0
2
,
n
−
2
T
λ
3
0
⋱
0
λ
n
]
,








{
More  generally,
∀
k
=
1
,
…
,
n
−
1
it  holds
a
k
b
k
+
c
k
d
k
=
a
k
c
k
+
b
k
d
k
=
0
a
k
2
+
b
k
2
=
c
k
2
+
d
k
2
=
a
k
2
+
c
k
2
=
b
k
2
+
d
k
2
=
1
}
,
G
3
=
(
1
⊕
P
2
⊕
I
n
−
3
)
G
2
(
1
⊕
P
2
T
⊕
I
n
−
3
)
=

[
1
0
0
0
a
2
b
2
0
c
2
d
2
0
3
,
n
−
3
0
3
,
n
−
3
T
I
n
−
3
]



[
λ
1
a
1
2
+
λ
2
b
1
2
(
λ
1
−
λ
2
)
a
1
c
1
(
λ
1
−
λ
2
)
a
1
c
1
λ
1
c
1
2
+
λ
2
d
1
2
0
2
,
n
−
2
0
2
,
n
−
2
T
λ
3
0
⋱
0
λ
n
]



[

1
0
0

0
a
2
c
2

0
b
2
d
2
0
3
,
n
−
3
0
3
,
n
−
3
T
I
n
−
3
]
=

[
λ
1
a
1
2
+
λ
2
b
1
2
(
λ
1
−
λ
2
)
a
1
a
2
c
1
(
λ
1
−
λ
2
)
a
1
c
1
c
2
(
λ
1
−
λ
2
)
a
1
a
2
c
1
(
λ
1
c
1
2
+
λ
2
d
1
2
)
a
2
2
+
λ
3
b
2
2
(
λ
1
c
1
2
+
λ
2
d
1
2
-
λ
3
)
a
2
c
2
(
λ
1
−
λ
2
)
a
1
c
1
c
2
(
λ
1
c
1
2
+
λ
2
d
1
2
-
λ
3
)
a
2
c
2
(
λ
1
c
1
2
+
λ
2
d
1
2
)
c
2
2
+
λ
3
d
2
2
0
3
,
n
−
3
0
3
,
n
−
3
T
λ
4
0
⋱
0
λ
n
]
.
Let λ
1
∗
:=
λ
1, ε
1
:
=
(
λ
1
∗
−
λ
2
)
a
1, and x
2
:=
c
1
ε
1, and let us firstly introduce
in (2.5) the following notation:
(2.6)
M
2
:
=
λ
1
a
1
2
+
λ
2
b
1
2
,


λ
2
∗
:
=
λ
1
*
c
1
2
+
λ
2
d
1
2
,


D
2
∗
:
=
diag
(
λ
2
∗
,
λ
3
)
,


D
2
:
=
diag
(
λ
4
,
…
,
λ
n
)
,
A
2
:
=
[
x
2
∣
0
]
T
,


M
3
:
=
[
λ
1
a
1
2
+
λ
2
b
1
2
(
λ
1
−
λ
2
)
a
1
a
2
c
1
(
λ
1
−
λ
2
)
a
1
a
2
c
1
(
λ
1
c
1
2
+
λ
2
d
1
2
)
a
2
2
+
λ
3
b
2
2
]
,
λ
3
∗
:
=
(
λ
1
c
1
2
+
λ
2
d
1
2
)
c
2
2
+
λ
3
d
2
2
,


D
3
∗
:
=
diag
(
λ
3
∗
,
λ
4
)
,
D
3
:
=
diag
(
λ
5
,
…
,
λ
n
)
,


A
3
:
=
[
(
λ
1
−
λ
2
)
a
1
c
1
c
2
(
λ
1
c
1
2
+
λ
2
d
1
2
−
λ
3
)
a
2
c
2
0
0
]
.
Thereafter, observing
the partition of G
2 and G
3 obtained in (2.5)
(2.7)
G
2
=
[
M
2
A
2
T
0
1
,
n
−
3
A
2
D
2
∗
0
2
,
n
−
3
0
1
,
n
−
3
T
0
2
,
n
−
3
T
D
2
]
,


G
3
=
[
M
3
A
3
T
0
2
,
n
−
4
A
3
D
3
∗
0
2
,
n
−
4
0
2
,
n
−
4
T
0
2
,
n
−
4
T
D
3
]
,
it can readily
be anticipated the partition of subsequent matrices G
k
(
k
=
4
,
…
,
n
−
2
) as follows:
(2.8)
G
k
=
[
M
k
A
k
T
0
k
−
1
,
n
−
k
−
1
A
k
D
k
∗
0
2
,
n
−
k
−
1
0
k
−
1
,
n
−
k
−
1
T
0
2
,
n
−
k
−
1
T
D
k
]
,


x
k
:
=
[
x
k
,
1
x
k
,
2
⋯
x
k
,
k
−
1
]
,

A
k
:
=
[
x
k
0
1
,
k
−
1
]
,
where M
k is the symmetric (
k
−
1
)
×
(
k
−
1
)
matrix, x
k is 1
×
(
k
−
1
)
row vector, A
k is 2
×
(
k
−
1
)
matrix, λ
k
∗ is modified eigenvalue λ
k, D
k
∗
:=
diag(
λ
k
∗
,
λ
k
+
1
) and D
k
:=
diag(
λ
k
+2
,
…
,
λ
n
)
. For k
=
2
,
…
,
n
−
3 from (2.1)–(2.3), (2.8) it follows
that
(2.9)
G
k
+
1
=
(
I
k
−
1
⊕
P
k
⊕
I
n
−
k
−
1
)
G
k
(
I
k
−
1
⊕
P
k
T
⊕
I
n
−
k
−
1
)
=
[
M
k
A
k
T
P
k
T
0
k
−
1
,
n
−
k
−
1
P
k
A
k
P
k
D
k
∗
P
k
T
0
2
,
n
−
k
−
1
0
k
−
1
,
n
−
k
−
1
T
0
2
,
n
−
k
−
1
T
D
k
]
,
P
k
A
k
=
[
a
k
x
k
,
1
⋯
a
k
x
k
,
k
−
1
c
k
x
k
,
1
⋯
c
k
x
k
,
k
−
1
]
,


P
k
D
k
∗
P
k
T
=
[
λ
k
∗
a
k
2
+
λ
k
+
1
b
k
2
(
λ
k
∗
−
λ
k
+
1
)
a
k
c
k
(
λ
k
∗
−
λ
k
+
1
)
a
k
c
k
λ
k
∗
c
k
2
+
λ
k
+
1
d
k
2
]
.
For k
=
2
,
…
,
n
−
3, let us define: λ
k
+
1
∗
:=
λ
k
∗
c
k
2
+
λ
k
+
1
d
k
2,  ε
k
:
=
(
λ
k
∗
−
λ
k
+
1
)
a
k,  ψ
k
k
:=
λ
k
∗
a
k
2
+
λ
k
+
1
b
k
2 and thereafter D
k
+
1
:=
diag(
λ
k
+3
,
…
,
λ
n
) and D
k
+
1
∗
:=
diag(
λ
k
∗
c
k
2
+
λ
k
+
1
d
k
2
,
λ
k
+
2
)
. Then, from (2.8)-(2.9) it follows the identification
(2.10)
M
k
+
1
:
=
[
M
k
a
k
x
k
T
a
k
x
k
ψ
k
k
]
,


A
k
+
1
=
[
x
k
+
1
0
1
,
k
]
:
=
[
c
k
x
k
c
k
ε
k
0
1
,
k
0
]
=
[
c
k
x
k
,
1
⋯
c
k
x
k
,
k
−
1
c
k
ε
k
0
⋯
0
0
],
which enables
the partition of (2.10)
G
k
+
1 in () to be like that of (2.10)
G
k in (), and that partition of (2.10)
x
k
+
1 be
rather simple
(2.10)
G
k
+
1
=
[
M
k
+
1
A
k
+
1
T
0
k
,
n
−
k
−
2
A
k
+
1
D
k
+
1
∗
0
2
,
n
−
k
−
2
0
k
,
n
−
k
−
2
T
0
2
,
n
−
k
−
2
T
D
k
+
1
]
,

x
k
+
1
:
=
c
k
[
x
k
∣
ε
k
]
,

k
=
2
,
…
,
n
−
3.
Let ψ
11
:
=
M
2. Having uncovered the partition
pattern of M
k
+
1
(
k
=
2
,
…
,
n
−
3
), we can pursue partitioning of M
n
−
2 backwardly from M
n
−
2 to M
2, by using (2.10)). Afterwards,
we can produce G
n
−
2, by using (2.10)-(2.11). The results are
(2.12)
M
n
−
2
=
[
ψ
11
a
2
x
2
T
a
2
x
2
ψ
22
a
3
x
3
T
a
3
x
3
ψ
33
⋯
⋮
⋱
a
n
−
3
x
n
−
3
T
a
n
−
3
x
n
−
3
ψ
n
−
3
,
n
−
3
]
,


G
n
−
2
=
[
M
n
−
2
x
n
−
2
T
x
n
−
2
λ
n
−
2
∗
0
n
−
2
,
1
0
1
,
n
−
2
λ
n
−
1
0
n
−
1
,
1
0
1
,
n
−
1
λ
n
]
.
Since G
n
−
1
:=
(
I
n
−
3
⊕
P
n
−
2
⊕
1
)
G
n
−
2
(
I
n
−
3
⊕
P
n
−
2
T
⊕
1
) and
(2.13)
P
n
−
2
[
λ
n
−
2
∗
0
0
λ
n
−
1
]
P
n
−
2
T
=
[
λ
n
−
2
∗
a
n
−
2
2
+
λ
n
−
1
b
n
−
2
2
(
λ
n
−
2
∗
−
λ
n
−
1
)
a
n
−
2
c
n
−
2
(
λ
n
−
2
∗
−
λ
n
−
1
)
a
n
−
2
c
n
−
2
λ
n
−
2
∗
c
n
−
2
2
+
λ
n
−
1
d
n
−
2
2
]
,
then after defining ψ
n
−
2
,
n
−
2
:=
λ
n
−
2
∗
a
n
−
2
2
+
λ
n
−
1
b
n
−
2
2
,
ε
n
−
2
:=
(
λ
n
−
2
∗
−
λ
n
−
1
)
a
n
−
2
,
λ
n
−
1
∗
:=
λ
n
−
2
∗
c
n
−
2
2
+
λ
n
−
1
d
n
−
2
2 and x
n
−
1
:=
c
n
−
2
[
x
n
−
2
∣
ε
n
−
2
]
, it follows from (2.12)-(2.13)
(2.14)
G
n
−
1
=
[
ψ
11
a
2
x
2
T
a
2
x
2
ψ
22
a
3
x
3
T
a
3
x
3
ψ
33
⋯
⋮
⋱
a
n
−
3
x
n
−
3
T
a
n
−
3
x
n
−
3
ψ
n
−
3
,
n
−
3
a
n
−
2
x
n
−
2
T
a
n
−
2
x
n
−
2
ψ
n
−
2
,
n
−
2
x
n
−
1
T
x
n
−
1
λ
n
−
1
∗
0
n
−
1
,
1
0
1
,
n
−
1
λ
n
]
.
Since G
n
:=
(
I
n
−
2
⊕
P
n
−
1
)
G
n
−
1
(
I
n
−
2
⊕
P
n
−
1
T
) and
(2.15)
P
n
−
1
[
λ
n
−
1
∗
0
0
λ
n
]
P
n
−
1
T
=
[
λ
n
−
1
∗
a
n
−
1
2
+
λ
n
b
n
−
1
2
(
λ
n
−
1
∗
−
λ
n
)
a
n
−
1
c
n
−
1
(
λ
n
−
1
∗
−
λ
n
)
a
n
−
1
c
n
−
1
λ
n
−
1
∗
c
n
−
1
2
+
λ
n
d
n
−
1
2
]
,
then on introducing ψ
n
−
1
,
n
−
1
:=
λ
n
−
1
∗
a
n
−
1
2
+
λ
n
b
n
−
1
2
,
ε
n
−
1
:=
(
λ
n
−
1
∗
−
λ
n
)
a
n
−
1, and λ
n
∗
:=
λ
n
−
1
∗
c
n
−
1
2
+
λ
n
d
n
−
1
2
, we obtain from (2.14)-(2.15) the partition of G
n which is amenable to the production of its entries in explicit form and is suitable
for further discussion about solving some specific IEPs
(2.16)
G
n
=
[
ψ
11
a
2
x
2
T
a
2
x
2
ψ
22
a
3
x
3
T
a
3
x
3
ψ
33
⋯
⋮
⋱
a
n
−
2
x
n
−
2
T
a
n
−
2
x
n
−
2
ψ
n
−
2
,
n
−
2
a
n
−
1
x
n
−
1
T
c
n
−
1
x
n
−
1
T
a
n
−
1
x
n
−
1
ψ
n
−
1
,
n
−
1
c
n
−
1
ε
n
−
1
c
n
−
1
x
n
−
1
c
n
−
1
ε
n
−
1
λ
n
∗
]
.
For k
=
2
,
…
,
n, we consecutively obtain from λ
1
∗
:=
λ
1 and λ
k
∗
:=
λ
k
−
1
∗
c
k
−
1
2
+
λ
k
d
k
−
1
2 that generally it holds
(2.17)
λ
k
∗
=
(
c
1
c
2
⋯
c
k
−
1
)
2
λ
1
+
(
d
1
c
2
⋯
c
k
−
1
)
2
λ
2
+
⋯
+
(
d
k
−
2
c
k
−
1
)
2
λ
k
−
1
+
d
k
−
1
2
λ
k
,

k
=
2
,
…
,
n.
Since (2.17)
ψ
11
=
λ
1
a
1
2
+
λ
2
b
1
2 and (2.17)
ψ
k
k
:
=
λ
k
∗
a
k
2
+
λ
k
+
1
b
k
2
(
k
=
2
,
…
,
n
−
1
), then from () it follows that
(2.17)
ψ
k
k
=
(
c
1
c
2
⋯
c
k
−
1
a
k
)
2
λ
1
+
(
d
1
c
2
⋯
c
k
−
1
a
k
)
2
λ
2
+
⋯
+
(
d
k
−
2
c
k
−
1
a
k
)
2
λ
k
−
1

+
(
d
k
−
1
a
k
)
2
λ
k
+
b
k
2
λ
k
+
1
,

k
=
2
,
…
,
n
−
1.
Observe that it
is not necessary to calculate “ψ”s from (2.18), but only the modified
eigenvalues from (2.17) since it holds ε
k
=
(
λ
k
∗
−
λ
k
+
1
)
a
k and ψ
k
k
=
λ
k
∗
a
k
2
+
λ
k
+
1
b
k
2
=
(
λ
k
∗
−
λ
k
+
1
)
a
k
2
+
(
a
k
2
+
b
k
2
)
λ
k
+
1
=
a
k
ε
k
+
λ
k
+
1
(
k
=
1
,
…
,
n
−
1
). As it is x
2
=
c
1
ε
1, then for k
=
2
,
…
,
n
−
2 from (2.10)) it follows that
(2.19)
x
k
+
1
=
c
k
[
x
k
∣
ε
k
]
=
c
k
[
c
k
−
1
[
x
k
−
1
∣
ε
k
−
1
]
∣
ε
k
]
=
[
c
k
c
k
−
1
x
k
−
1
∣
c
k
c
k
−
1
ε
k
−
1
∣
c
k
ε
k
]
=
⋯
=
[
c
k
c
k
−
1
⋯
c
2
x
2
∣
c
k
c
k
−
1
⋯
c
2
ε
2
∣
⋯
∣
c
k
c
k
−
1
ε
k
−
1
∣
c
k
ε
k
]
=
[
c
k
c
k
−
1
⋯
c
2
c
1
ε
1
∣
c
k
c
k
−
1
⋯
c
2
ε
2
∣
⋯
∣
c
k
c
k
−
1
ε
k
−
1
∣
c
k
ε
k
]
(2.20)
a
k
x
k
=
[
a
k
c
k
−
1
c
k
−
2
⋯
c
2
c
1
ε
1
∣
a
k
c
k
−
1
c
k
−
2
⋯
c
2
ε
2
∣
⋯
∣
a
k
c
k
−
1
c
k
−
2
ε
k
−
2
∣
a
k
c
k
−
1
ε
k
−
1
]
,
k
=
2
,
…
,
n
−
1.

The real
symmetric matrix G
n with assigned spectrum {
λ
1
,
λ
2
,
…
,
λ
n
} and the
explicitly expressed entries can be derived from (2.16) and (2.20), bearing on mind
that “ψ”s and “ε”s are calculated by using {
λ
1
,
λ
2
,
…
,
λ
n
}, P
k,
modified eigenvalues (2.17) and ε
k
=
(
λ
k
∗
−
λ
k
+
1
)
a
k
(
k
=
1
,
…
,
n
−
1
):
(2.21)
G
n
=
[
ψ
11
a
2
c
1
ε
1
a
3
c
2
c
1
ε
1
a
4
c
3
c
2
c
1
ε
1
⋯
P
S
a
2
c
1
ε
1
ψ
22
a
3
c
2
ε
2
a
4
c
3
c
2
ε
2
⋯
U
F
a
3
c
2
c
1
ε
1
a
3
c
2
ε
2
⋮
⋮
⋯
⋮
⋮
a
4
c
3
c
2
c
1
ε
1
a
4
c
3
c
2
ε
2
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋱
⋮
⋮
P
U
⋯
⋯
⋯
ψ
n
−
1
,
n
−
1
c
n
−
1
ε
n
−
1
S
F
⋯
⋯
⋯
c
n
−
1
ε
n
−
1
λ
n
∗
]
,
where 𝒫 denotes a
n
−
1
c
n
−
2
⋯
c
2
c
1
ε
1, 𝒮 denotes c
n
−
1
c
n
−
2
⋯
c
2
c
1
ε
1, 𝒰 denotes a
n
−
1
c
n
−
2
⋯
c
2
ε
2, and F denotes c
n
−
1
c
n
−
2
⋯
c
2
ε
2. Entries  of G
n
=
G
n
T
=
[
g
k
m
]
n
×
n are g
k
k
=
ψ
k
k
(
k
=
1
,
…
,
n
−
1
), g
n
n
=
λ
n
∗, g
k
m
=
a
k
c
k
−
1
c
k
−
2
⋯
c
m
ε
m
(
k
>
m
;
k
=
2
,
…
,
n
−
1
) and g
n
m
=
c
n
−
1
c
n
−
2
⋯
c
m
ε
m
(
m
=
1
,
…
,
n
−
1
). They are calculated according to the following steps:(a)
Select arbitrarily the entries{
a
k
,
b
k
,
c
k
,
d
k
} of 2
×
2 orthogonal matrices P
k
(
k
=
1
,
…
,
n
−
1
), given by (2.1);
(b)
withλ
1
∗
:=
λ
1,
calculate the modified eigenvalues λ
k
∗
(
k
=
2
,
…
,
n
)
, by using (2.17);
(c)
calculateε
k
=
(
λ
k
∗
−
λ
k
+
1
)
a
k and ψ
k
k
=
a
k
ε
k
+
λ
k
+
1
(
k
=
1
,
…
,
n
−
1
);
(d)
calculate the entries ofG
n,
by using (2.21).MatrixU (2.4) is n
×
n orthogonal
modal matrix established from
eigenvectors of G
n. We will now prove that U is not only orthogonal, but also lower
Hessenberg with explicitly expressed entries. Let us firstly produce U
1
T
U
2
T and U
1
T
U
2
T
U
3
T
, whose partition will enable us to anticipate the
partition of U
1
T
U
2
T
U
3
T
⋯
U
k
T
(
k
=
4
,
…
,
n
−
1
)
(2.22)
U
1
T
U
2
T
=
[
a
1
a
2
c
1
c
2
c
1
b
1
a
2
d
1
c
2
d
1
0
b
2
d
2
0
3
,
n
−
3
0
3
,
n
−
3
T
I
n
−
3
]
,


U
1
T
U
2
T
U
3
T
=
[
a
1
a
2
c
1
a
3
c
2
c
1
c
3
c
2
c
1
b
1
a
2
d
1
a
3
c
2
d
1
c
3
c
2
d
1
0
b
2
a
3
d
2
c
3
d
2
0
0
b
3
d
3
0
4
,
n
−
4
0
4
,
n
−
4
T
I
n
−
4
]
.
If we now suppose
that U
1
T
U
2
T
⋯
U
k
T
:=
H
k
+
1
⊕
I
n
−
k
−
1
(
k
=
2
,
…
,
n
−
1
) where H
k
+
1 is orthogonal (
k
+
1
)
×
(
k
+
1
)
upper Hessenberg matrix
(2.23)
H
k
+
1
=
[
a
1
a
2
c
1
a
3
c
2
c
1
⋯
a
k
c
k
−
1
⋯
c
2
c
1
c
k
c
k
−
1
⋯
c
2
c
1
b
1
a
2
d
1
a
3
c
2
d
1
⋯
a
k
c
k
−
1
⋯
c
2
d
1
c
k
c
k
−
1
⋯
c
2
d
1
0
b
2
a
3
d
2
⋯
a
k
c
k
−
1
⋯
c
3
d
2
c
k
c
k
−
1
⋯
c
3
d
2
⋯
⋯
⋯
⋯
⋯
⋯
0
⋯
0
b
k
−
1
a
k
d
k
−
1
c
k
d
k
−
1
0
⋯
⋯
0
b
k
d
k
]
,
then since according
to (2.2), it holds U
k
+
1
:=
I
k
⊕
P
k
+
1
⊕
I
n
−
k
−
2
, we may write further for k
=
2
,
…
,
n
−
2
(2.24)
U
1
T
U
2
T
⋯
U
k
T
U
k
+
1
T
=
(
H
k
+
1
⊕
I
n
−
k
−
1
)
U
k
+
1
T
=
(
H
k
+
1
⊕
1
⊕
I
n
−
k
−
2
)
(
I
k
⊕
P
k
+
1
T
⊕
I
n
−
k
−
2
)
=
[
(
H
k
+
1
⊕
1
)
(
I
k
⊕
P
k
+
1
T
)
]
⊕
I
n
−
k
−
2
=
H
k
+
2
⊕
I
n
−
k
−
2
,

where
H
k
+
2
:
=
(
H
k
+
1
⊕
1
)
(
I
k
⊕
P
k
+
1
T
)
.
By using (2.2), (2.23)-(2.24), it follows that
(2.25)
H
k
+
2
=
[
a
1
a
2
c
1
a
3
c
2
c
1
⋯
a
k
c
k
−
1
⋯
c
2
c
1
c
k
c
k
−
1
⋯
c
2
c
1
0
b
1
a
2
d
1
a
3
c
2
d
1
⋯
a
k
c
k
−
1
⋯
c
2
d
1
c
k
c
k
−
1
⋯
c
2
d
1
0
0
b
2
a
3
d
2
⋯
a
k
c
k
−
1
⋯
c
3
d
2
c
k
c
k
−
1
⋯
c
3
d
2
0
⋯
⋯
⋯
⋯
⋯
⋯
⋯
0
⋯
0
b
k
−
1
a
k
d
k
−
1
c
k
d
k
−
1
0
0
⋯
⋯
0
b
k
d
k
0
0
⋯
⋯
0
0
0
1
]
[
I
k
0
k
,
2
0
k
,
2
T
a
k
+
1
c
k
+
1
b
k
+
1
d
k
+
1
]
=
[
a
1
a
2
c
1
a
3
c
2
c
1
⋯
a
k
c
k
−
1
⋯
c
2
c
1
a
k
+
1
c
k
c
k
−
1
⋯
c
2
c
1
c
k
+
1
c
k
c
k
−
1
⋯
c
2
c
1
b
1
a
2
d
1
a
3
c
2
d
1
⋯
a
k
c
k
−
1
⋯
c
2
d
1
a
k
+
1
c
k
c
k
−
1
⋯
c
2
d
1
c
k
+
1
c
k
c
k
−
1
⋯
c
2
d
1
0
b
2
a
3
d
2
⋯
a
k
c
k
−
1
⋯
c
3
d
2
a
k
+
1
c
k
c
k
−
1
⋯
c
3
d
2
c
k
+
1
c
k
c
k
−
1
⋯
c
3
d
2
⋯
⋯
⋯
⋯
⋯
⋯
⋯
0
⋯
0
b
k
−
1
a
k
d
k
−
1
a
k
+
1
c
k
d
k
−
1
c
k
+
1
c
k
d
k
−
1
0
⋯
⋯
0
b
k
a
k
+
1
d
k
c
k
+
1
d
k
0
⋯
⋯
0
0
b
k
+
1
d
k
+
1
]
,
and thereby it
is proved our previous assumption that U
1
T
U
2
T
⋯
U
k
T
:=
H
k
+
1
⊕
I
n
−
k
−
1
(
k
=
2
,
…
,
n
−
1
), where H
k
+
1 (2.23) is the orthogonal upper Hessenberg (
k
+
1
)
×
(
k
+
1
)
matrix with entries expressed explicitly. And finally, for k
=
n
−
1 from U
1
T
U
2
T
⋯
U
k
T
:=
H
k
+
1
⊕
I
n
−
k
−
1 and (2.4), (2.23), we obtain H
n
=
U
T
=
U
1
T
U
2
T
⋯
U
n
−
1
T and
(2.26)


U
=
H
n
T
=
U
n
−
1
U
n
−
2
⋯
U
1
=

[
a
1
b
1
0
0
⋯
0
0
a
2
c
1
a
2
d
1
b
2
0
⋯
0
0
a
3
c
2
c
1
a
3
c
2
d
1
a
3
d
2
⋮
⋯
⋮
⋮
a
4
c
3
c
2
c
1
a
4
c
3
c
2
d
1
a
4
c
3
d
2
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋱
⋮
⋮
a
n
−
1
c
n
−
2
⋯
c
2
c
1
a
n
−
1
c
n
−
2
⋯
c
2
d
1
⋯
⋯
⋯
a
n
−
1
d
n
−
2
b
n
−
1
c
n
−
1
c
n
−
2
⋯
c
2
c
1
c
n
−
1
c
n
−
2
⋯
c
2
d
1
⋯
⋯
⋯
c
n
−
1
d
n
−
2
d
n
−
1
]

.
The entries
of the orthogonal lower Hessenberg matrix U
=
[
u
k
m
]
(
k
,
m
=
1
,
…
,
n
) are defined as follows:
(2.27)
u
k
m
=
0

(
m
>
k
+
1
;
k
,
m
=
1
,
…
,
n
)
,


u
k
,
k
+
1
=
b
k

(
k
=
1
,
…
,
n
−
1
),


u
11
=
a
1
,
u
k
k
=
a
k
d
k
−
1

(
k
=
2
,
…
,
n
−
1
)
,


u
k
,
1
=
a
k
c
k
−
1
c
k
−
2
⋯
c
2
c
1

(
k
=
2
,
…
,
n
−
1
)
,
u
n
,
1
=
c
n
−
1
c
n
−
2
⋯
c
1
,


u
n
,
k
=
c
n
−
1
c
n
−
2
⋯
c
k
d
k
−
1

(
k
=
2
,
…
,
n
−
1
)
,
u
k
m
=
a
k
c
k
−
1
⋯
c
m
d
m
−
1

(
m
+
1
≤
k
≤
n
−
1
;
m
=
2
,
…
,
n
−
1
)
.
By using the similar arguments as in derivation of entries of matrix U, the orthogonal matrix V which is to be produced by using
(2.4) can be shown to take
on upper Hessenberg form. Proving of this fact goes with similar paces that
were used for obtaining U and it is
left to the reader.


## 3. The Explicit Solution of the IEP of Real Symmetric Matrices with Some Specific Sign Patterns

Let the real
eigenvalues from the spectrum{
λ
1
,
λ
2
,
…
,
λ
n
} be arbitrarily enumerated, thereby establishing
the sequence {
λ
k
}
(
k
=
1
,
…
,
n
). The nonnegative sequence will be denoted by {
λ
k
}
≥
0,
and the nonpositive one by {
λ
k
}
≤
0
(
k
=
1
,
…
,
n
)
. Firstly, we will prove two lemmas.Lemma 3.1.
If
the sequence{
λ
k
}
≥
0
(
k
=
1
,
…
,
n
) is increasing [decreasing], then in (2.21) λ
n
∗
≥
0
,
ψ
m
m
≥
0
, and the sequence  {
a
m
ε
m
}
≤
0
[
{
a
m
ε
m
}
≥
0
]
(
m
=
1
,
…
,
n
−
1
).Proof.
Since{
λ
k
}
≥
0
(
k
=
1
,
…
,
n
),
then it is trivial to see from (2.17) and (2.18) that all diagonal entries of G
n are nonnegative, that is, λ
n
∗
≥
0 and ψ
m
m
≥
0
(
m
=
1
,
…
,
n
−
1
) no matter whether the sequence {
λ
k
}
≥
0 is increasing or decreasing. By virtue of orthogonality
of P
k,
we have c
k
2
+
d
k
2
=
1
(
k
=
1
,
…
,
n
).
If {
λ
k
}
(
k
=
1
,
…
,
n
) is increasing sequence, then for m
=
1 we have a
1
ε
1
=
(
λ
1
∗
−
λ
2
)
a
1
2
=
(
λ
1
−
λ
2
)
a
1
2
≤
0 and for m
=
2
,
…
,
n
−
1 we obtain (3.1)
d
m
−
1
2
λ
m
−
λ
m
+
1

≤
d
m
−
1
2
λ
m
−
λ
m
=
λ
m
(
1
−
d
m
−
1
2
)
=
−
c
m
−
1
2
λ
m
≤
−
c
m
−
1
2
λ
m
−
1
,
(
d
m
−
2
c
m
−
1
)
2
λ
m
−
1
+
d
m
−
1
2
λ
m
−
λ
m
+
1

≤
(
d
m
−
2
c
m
−
1
)
2
λ
m
−
1
−
c
m
−
1
2
λ
m
−
1
=
−
(
c
m
−
2
c
m
−
1
)
2
λ
m
−
1
≤
−
(
c
m
−
2
c
m
−
1
)
2
λ
m
−
2
,
(
d
m
−
3
c
m
−
2
c
m
−
1
)
2
λ
m
−
2
+
(
d
m
−
2
c
m
−
1
)
2
λ
m
−
1
+
d
m
−
1
2
λ
m
−
λ
m
+
1

≤
−
(
c
m
−
3
c
m
−
2
c
m
−
1
)
2
λ
m
−
2
≤
−
(
c
m
−
3
c
m
−
2
c
m
−
1
)
2
λ
m
−
3
,
(
d
1
c
2
⋯
c
m
−
1
)
2
λ
2
+
⋯
+
(
d
m
−
2
c
m
−
1
)
2
λ
m
−
1
+
d
m
−
1
2
λ
m
−
λ
m
+
1

≤
−
(
c
1
⋯
c
m
−
1
)
2
λ
2
≤
−
(
c
1
⋯
c
m
−
1
)
2
λ
1
. From (2.17) and
the last of inequalities (3.1) it follows λ
m
∗
≤
λ
m
+
1 and a
m
ε
m
=
(
λ
m
∗
−
λ
m
+
1
)
a
m
2
≤
0
(
m
=
2
,
…
,
n
−
1
). If {
λ
k
}
(
k
=
1
,
…
,
n
) is decreasing sequence, then for m
=
1 we have a
1
ε
1
=
(
λ
1
∗
−
λ
2
)
a
1
2
=
(
λ
1
−
λ
2
)
a
1
2
≥
0 and for m
=
2
,
…
,
n
−
1 we obtain (3.2)
d
m
−
1
2
λ
m
−
λ
m
+
1

≥
d
m
−
1
2
λ
m
−
λ
m
=
−
λ
m
(
1
−
d
m
−
1
2
)
=
−
c
m
−
1
2
≥
−
c
m
−
1
2
λ
m
−
1
,
(
d
m
−
2
c
m
−
1
)
2
λ
m
−
1
+
d
m
−
1
2
λ
m
−
λ
m
+
1

≥
(
d
m
−
2
c
m
−
1
)
2
λ
m
−
1
−
c
m
−
1
2
λ
m
−
1
=
−
(
c
m
−
2
c
m
−
1
)
2
λ
m
−
1
≥
−
(
c
m
−
2
c
m
−
1
)
2
λ
m
−
2
,
(
d
m
−
3
c
m
−
2
c
m
−
1
)
2
λ
m
−
2
+
(
d
m
−
2
c
m
−
1
)
2
λ
m
−
1
+
d
m
−
1
2
λ
m
−
λ
m
+
1

≥
−
(
c
m
−
3
c
m
−
2
c
m
−
1
)
2
λ
m
−
2
≥
−
(
c
m
−
3
c
m
−
2
c
m
−
1
)
2
λ
m
−
3
,
(
d
1
c
2
⋯
c
m
−
1
)
2
λ
2
+
⋯
+
(
d
m
−
2
c
m
−
1
)
2
λ
m
−
1
+
d
m
−
1
2
λ
m
−
λ
m
+
1

≥
−
(
c
1
⋯
c
m
−
1
)
2
λ
2
≥
−
(
c
1
⋯
c
m
−
1
)
2
λ
1
. From (2.17) and
the last of inequalities (3.2) it follows λ
m
∗
≥
λ
m
+
1 and a
m
ε
m
=
(
λ
m
∗
−
λ
m
+
1
)
a
m
2
≥
0
(
m
=
2
,
…
,
n
−
1
). This
completes the proof of lemma. For a nonpositive sequence, an analogous lemma can be formulated.Lemma 3.2.
If
the sequence{
λ
k
}
≤
0
(
k
=
1
,
…
,
n
)
is
increasing [decreasing], then in (2.21) λ
n
∗
≤
0
,
ψ
m
m
≤
0 and the sequence  {
a
m
ε
m
}
≤
0
[
{
a
m
ε
m
}
≥
0
]
(
m
=
1
,
…
,
n
−
1
).Proof.
It is similar to that of Lemma3.1, but
in this case the diagonal entries of G
n are nonpositive, that is, λ
n
∗
≤
0 and ψ
m
m
≤
0
(
m
=
1
,
…
,
n
−
1
), no matter whether the sequence {
λ
k
}
≤
0 is increasing or decreasing (see (2.18)).Now, we shall
formulate a new theorem related to explicit solving of IEP of real symmetric
matrices with some specific sign patterns.Theorem 3.3.
Ifθ
k
(
k
=
1
,
…
,
n
−
1
)
are
arbitrarily selected angles from the range  [
0
,
π
/
2
], then the entries of real symmetric
matrices G
n with assigned spectrum  {
λ
1
,
λ
2
,
…
,
λ
n
}, produced by (2.21), can attain the following twelve sign patterns (zero entries are permitted), depending
on selection of matrices P
k
(
k
=
1
,
…
,
n
−
1
) (see (2.1)).Case 1.
(3.3)
P
k
=
A
k
=
[
cos
θ
k
−
sin
θ
k
sin
θ
k
cos
θ
k
]
,

or

P
k
=
B
k
=
[
cos
θ
k
sin
θ
k
sin
θ
k
−
cos
θ
k
]

G
n
=
[
+
+
+
⋱
+
+
]
,

λ
1
≥
λ
2
≥
⋯
≥
λ
n
≥
0


G
n
=
[
−
−
+
⋱
+
−
]
,

λ
n
≤
λ
n
−
1
≤
⋯
≤
λ
1
≤
0
G
n
=
[
+
+
−
⋱
−
+
]
,

λ
n
≥
λ
n
−
1
≥
⋯
≥
λ
1
≥
0


G
n
=
[
−
−
−
⋱
−
−
]

λ
1
≤
λ
2
≤
⋯
≤
λ
n
≤
0
.Case 2.
(2)
P
k
=
C
k
=
[
cos
θ
k
sin
θ
k
−
sin
θ
k
cos
θ
k
]

G
n
=
[
+
−
+
⋯
(
−
1
)
n
−
1
−
+
−
⋯
(
−
1
)
n
−
2
⋯
⋯
⋯
⋯
⋯
(
−
1
)
n
−
2
⋯
⋯
+
−
(
−
1
)
n
−
1
⋯
⋯
−
+
]
,


λ
1
≥
λ
2
≥
⋯
≥
λ
n
≥
0


G
n
=
[
−
−
+
⋯
(
−
1
)
n
−
1
−
−
−
⋯
(
−
1
)
n
−
2
⋯
⋯
⋯
⋯
⋯
(
−
1
)
n
−
2
⋯
⋯
−
−
(
−
1
)
n
−
1
⋯
⋯
−
−
]
,


λ
n
≤
λ
n
−
1
≤
⋯
≤
λ
1
≤
0
G
n
=
[
+
+
−
⋯
(
−
1
)
n
+
+
+
⋯
(
−
1
)
n
−
1
⋯
⋯
⋯
⋯
⋯
(
−
1
)
n
−
1
⋯
⋯
+
+
(
−
1
)
n
⋯
⋯
+
+
]
,


λ
n
≥
λ
n
−
1
≥
⋯
≥
λ
1
≥
0


G
n
=
[
−
+
−
⋯
(
−
1
)
n
+
−
+
⋯
(
−
1
)
n
−
1
⋯
⋯
⋯
⋯
⋯
(
−
1
)
n
−
1
⋯
⋯
−
+
(
−
1
)
n
⋯
⋯
+
−
]


λ
1
≤
λ
2
≤
⋯
≤
λ
n
≤
0
.Case 3.
(3.5)
P
k
=
D
k
=
[
−
cos
θ
k
sin
θ
k
sin
θ
k
cos
θ
k
]

G
n
=
[
+
+
+
⋯
+
−
+
+
+
⋯
+
−
⋯
⋯
⋯
⋯
⋯
⋯
+
+
+
⋯
+
−
−
−
−
⋯
−
+
]
,


λ
1
≥
λ
2
≥
⋯
≥
λ
n
≥
0


G
n
=
[
−
+
+
⋯
+
−
+
−
+
⋯
+
−
⋯
⋯
⋯
⋯
⋯
⋯
+
+
⋯
⋯
−
−
−
−
⋯
⋯
−
−
]
,


λ
n
≤
λ
n
−
1
≤
⋯
≤
λ
1
≤
0
G
n
=
[
+
−
−
⋯
−
+
−
+
−
⋯
−
+
⋯
⋯
⋯
⋯
⋯
⋯
−
−
⋯
⋯
+
+
+
+
⋯
⋯
+
+
]
,


λ
n
≥
λ
n
−
1
≥
⋯
≥
λ
1
≥
0


G
n
=
[
−
−
−
⋯
−
+
−
−
−
⋯
−
+
⋯
⋯
⋯
⋯
⋯
⋯
−
−
⋯
⋯
−
+
+
+
⋯
⋯
+
+
]
.


λ
1
≤
λ
2
≤
⋯
≤
λ
n
≤
0Proof.
Ifθ
k
∈
[
0
,
π
/2],
then the signs of  a
k and c
k depend solely on selection of canonic orthogonal
matrices P
k
(
k
=
1
,
…
,
n
−
1
).
For any sign of sequence {
λ
m
}
(
m
=
1
,
…
,
n
) and its monotonicy realized through enumeration
of its members, one can readily check the sign patterns stated above: by using
(2.18) to determine signs of the diagonal entries in G
n and by
using Lemma 3.1 or Lemma 3.2 to determine signs of ε
k
(
k
=
1
,
…
,
n
−
1
).
Observe that only in Case 1 when λ
n
≥
λ
n
−
1
≥
⋯
≥
λ
1
≥
0
, that is, when the sequence {
λ
m
}
(
m
=
1
,
…
,
n
) is nonnegative and increasing (but not
strictly), matrix G
n is produced with hd sign pattern, including the possible
presence of zero entries. G
n may attain a sparse
structure if, for example, some eigenvalues are equal. To see that, let us firstly
suppose λ
1
=
⋯
=
λ
k
=
λ. Then from (2.17)-(2.18) it follows that λ
1
∗
=
⋯
=
λ
k
∗
=
λ
,
ψ
11
=
⋯
=
ψ
k
−
1
,
k
−
1
=
λ and ε
1
=
⋯
=
ε
k
−
1
=
0
, thus obviously making the matrix G
n (2.21) with sparse structure. By using (2.17)-(2.18), (2.21) and both Lemmas, we can
readily infer that if θ
k
∈
(
0
,
π
/2)
(
k
=
1
,
…
,
n
−
1
) and the
sequence {
λ
m
}
(
m
=
1
,
…
,
n
) is strictly
monotone, then matrix G
n (2.21) is produced with no zero
entries in all three considered cases.Remark 3.4.
Letλ
1
λ
2
⋯
λ
n
≠
0.
Then, since G
n
=
U
G
1
U
T and U
−
1
=
U
T (recall that U is orthogonal), it follows that G
n
−
1
=
(
U
T
)
−
1
G
1
−
1
U
−
1
=
U
G
1
−
1
U
T.
Also, when the sequence {
λ
m
}
(
m
=
1
,
…
,
n
) is increasing (decreasing), then the sequence {
λ
m
−
1
}
(
m
=
1
,
…
,
n
) is decreasing (increasing). These facts and Theorem
3.3 offer a possibility of determining the sign pattern of G
n
−
1 without really inverting G
n.
Furthermore, by using (2.17)-(2.18), (2.21), G
n
−
1 can be calculated explicitly, also without really
inverting G
n.Theorem 3.5.
Let
the positive increasing sequence{
λ
m
}
(
m
=
1
,
…
,
n
) be the
spectrum of G
n produced by using Case 1 of Theorem 3.3. Then there always exists such
a diagonal matrix D
:=
d
i
a
g
(
d
1
,
d
2
,
…
,
d
n
) with positive
diagonal entries which makes D
G
n
D truly hyperdominant.Proof.
IfG
1
:=
diag
(
λ
1
,
λ
2
,
…
,
λ
n
), then by Case 1 of Theorem 3.3, the nonsingular matrix G
n
=
U
G
1
U
T will have hd sign pattern and by Remark 3.4
G
n
−
1
=
U
G
1
−
1
U
T will be nonnegative matrix. Since d
m
>
0
(
m
=
1
,
…
,
n
), then the nonsingular symmetric matrix D
G
n
D is produced with hd sign pattern, but
it may not be truly hd, unless hd margin of each of its rows (or columns) is
nonnegative (recall that hd margin of a row or a column is sum of all entries in
that row or column). If G
n
=
[
g
k
m
]
(
k
,
m
=
1
,
…
,
n
), then hd margin p
k of the kth row (or the kth column) in D
G
n
D is given by (3.6)
p
k
=
∑
m
=
1
n
g
k
m
d
m
d
k
=
d
k
α
k
,

where
α
k
:
=
∑
m
=
1
n
g
k
m
d
m
,
k
=
1
,
…
,
n
. Let we arbitrarily
select α
k
>
0
(
k
=
1
,
…
,
n
) and let a
:=
[
α
1
α
2
⋯
α
n
]
T
α
1
α
2
⋯
α
n
]
T
,
col
(
D
)
:=
[
d
1
d
2
⋯
d
n
]
T
d
1
d
2
⋯
d
n
]
T and p
:=
[
p
1
p
2
⋯
p
n
]
T
p
1
p
2
⋯
p
n
]
T. Then, from (3.6) it follows that G
n
col
(
D
)
=
a, that is, col
(
D
)
=
G
n
−
1
a
>
0
n
,
1 and p
=
D
a
>
0
n
,
1. This not only
means that D
G
n
D has hd
sign pattern, but that it is truly hd furthermore. Obviously, as much as “α”s are assumed
greater, the greater will be row (column) hd margins of D
G
n
D. This completes the proof of theorem.


## 4. Explicit Solution of IEP of Hd Matrices with Uncommitted and with Assigned Nonnegative Spectrum

Theorem 4.1.
Letθ
k
(
k
=
1
,
…
,
n
−
1
) be a set of angles selected from the range [
0
,
π
/
2
] and let {
λ
1
,
λ
2
,
…
,
λ
n
} be uncommitted nonnegative spectrum of the real
symmetric matrix G
n
=
U
G
1
U
T
[
G
1
=
d
i
a
g
(
λ
1
,
λ
2
,
…
,
λ
n
)
] which is to
be produced as truly hd. Suppose that through enumeration of eigenvalues the sequence {
λ
m
}
≥
0
(
m
=
1
,
…
,
n
) is made increasing. Then, matrix G
n given by (2.21) will be truly hyperdominant if λ
1 is
sufficiently great.Proof.
Since by assumption the conditions
of Theorem3.3 (Case 1) are satisfied, then G
n produced by using (2.21) has hd
sign pattern. As it is ε
k
=
(
λ
k
∗
−
λ
k
+
1
)
a
k
(
k
=
1
,
…
,
n
−
1
),
then from (2.17)-(2.18), (2.21) it follows that hd margin p
m of the mth
row (or column) from G
n
(
m
=
1
,
…
,
n
) can be in
general represented as (4.1)
p
m
=
α
1
(
m
)
λ
1
+
α
2
(
m
)
λ
2
+
⋯
+
α
m
(
m
)
λ
m
+
α
m
+
1
(
m
)
λ
m
+
1

(
m
=
1
,
…
,
n
−
1
)
,
p
n
=
α
1
(
n
)
λ
1
+
α
2
(
n
)
λ
2
+
⋯
+
α
n
−
1
(
n
)
λ
n
−
1
+
α
n
(
n
)
λ
n
, where “α” coefficients
are defined as follows: (4.2)
α
1
(
1
)
:
=
a
1
(
a
1
+
a
2
c
1
+
a
3
c
2
c
1
+
⋯
+
a
n
−
1
c
n
−
2
⋯
c
2
c
1
+
c
n
−
1
c
n
−
2
⋯
c
2
c
1
)
,

m
=
1
,
α
2
(
1
)
:
=
b
1
(
b
1
+
a
2
d
1
+
a
3
c
2
d
1
+
⋯
+
a
n
−
1
c
n
−
2
⋯
c
2
d
1
+
c
n
−
1
c
n
−
2
⋯
c
2
d
1
)
,
α
1
(
m
)
:
=
(
a
m
c
m
−
1
⋯
c
2
c
1
)
(
a
1
+
a
2
c
1
+
a
3
c
2
c
1
+
⋯
+
a
m
c
m
−
1
⋯
c
2
c
1









+
⋯
+
a
n
−
1
c
n
−
2
⋯
c
2
c
1
+
c
n
−
1
c
n
−
2
⋯
c
2
c
1
)
,
α
2
(
m
)
:
=
(
a
m
c
m
−
1
⋯
c
2
d
1
)
(
b
1
+
a
2
d
1
+
a
3
c
2
c
1
+
⋯
+
a
m
c
m
−
1
⋯
c
2
d
1
+
⋯
+
a
n
−
1
c
n
−
2
⋯
c
2
d
1









+
c
n
−
1
c
n
−
2
⋯
c
2
d
1
)
,

p
=
3
,
…
,
m
,
α
p
(
m
)
:
=
(
a
m
c
m
−
1
⋯
c
p
d
p
−
1
)
(
b
p
−
1
+
a
p
d
p
−
1
+
a
p
+
1
c
p
d
p
−
1
+
⋯
+
a
n
−
1
c
n
−
2
⋯
c
p
d
p
−
1









+
c
n
−
1
c
n
−
2
⋯
c
p
d
p
−
1
)
,

m
=
2
,
…
,
n
−
1
,
p
=
3
,
…
,
m
α
m
(
m
)
:
=
(
a
m
d
m
−
1
)
(
b
m
−
1

+

a
m
d
m
−
1

+

a
m
+
1
c
m
d
m
−
1

+

⋯

+

a
n
−
1
c
n
−
2
⋯
c
m
d
m
−
1

+

c
n
−
1
c
n
−
2
⋯
c
m
d
m
−
1
)
,
α
m
+
1
(
m
)
:
=
b
m
(
b
m
+
a
m
+
1
d
m
+
a
m
+
2
c
m
+
1
d
m
+
⋯
+
a
n
−
1
c
n
−
2
⋯
c
m
+
1
d
m
+
c
n
−
1
c
n
−
2
⋯
c
m
+
1
d
m
)
,




























m
=
2
,
…
,
n
−
2
,
α
n
(
n
−
1
)
:
=
b
n
−
1
(
b
n
−
1
+
d
n
−
1
)
,
α
1
(
n
)
:
=
(
c
n
−
1
c
n
−
2
⋯
c
2
c
1
)
(
a
1
+
a
2
c
1
+
a
3
c
2
c
1
+
⋯
+
a
n
−
1
c
n
−
2
⋯
c
2
c
1
+
c
n
−
1
c
n
−
2
⋯
c
2
c
1
)
,
α
2
(
n
)
:
=
(
c
n
−
1
c
n
−
2
⋯
c
2
d
1
)
(
b
1
+
a
2
d
1
+
a
3
c
2
d
1
+
⋯
+
a
n
−
1
c
n
−
2
⋯
c
2
d
1
+
c
n
−
1
c
n
−
2
⋯
c
2
d
1
)
α
p
(
n
)
:
=
(
c
n
−
1
c
n
−
2
⋯
c
p
d
p
−
1
)
(
b
p
−
1
+
a
p
d
p
−
1
+
a
p
+
1
c
p
d
p
−
1
+
⋯
+
a
n
−
1
c
n
−
2
⋯
c
p
d
p
−
1










+
c
n
−
1
c
n
−
2
⋯
c
p
d
p
−
1
)
,

p
=
3
,
…
,
n
−
1.
α
n
(
n
)
:
=
d
n
−
1
(
b
n
−
1
+
d
n
−
1
)
. According to Case 1 of Theorem 3.3, both a
k and c
k are nonnegative when θ
k
∈
[
0
,
π
/2]
(
k
=
1
,
…
,
n
−
1
). Then, from
(4.2) we see that α
1
(
m
)
≥
0
(
m
=
1
,
…
,
n
),
whereas other “α”s may be nonpositive. Since “α”s depend only
on selection of “θ”s, then by presuming λ
1
=
λ
2
=
⋯
=
λ
n
=
λ
≠
0, we obtain from (2.21) G
n
=
λ
I
n and p
m
=
λ
(
m
=
1
,
…
,
n
) and from (4.1) we conclude that in general it holds: (4.3)
∑
p
=
1
m
+
1
α
p
(
m
)
≡
1

(
m
=
1
,
…
,
n
−
1
),


∑
q
=
1
n
α
q
(
n
)
≡
1. Although G
n is produced with hd sign pattern, it will not be truly hd unless each of its
row (column) hd margins is nonnegative, that is, p
=
[
p
1
p
2
⋯
p
n
]
T
p
1
p
2
⋯
p
n
]
T
≥
0
n
,
1.
The column vector p with entries (4.1)
can be written as (4.4)
p
=
[
α
1
(
1
)
α
2
(
1
)
0
0
⋯
0
α
1
(
2
)
α
2
(
2
)
α
3
(
2
)
0
⋯
0
⋯
⋯
⋯
⋯
⋯
⋯
α
1
(
n
−
1
)
α
2
(
n
−
1
)
α
3
(
n
−
1
)
⋯
⋯
α
n
(
n
−
1
)
α
1
(
n
)
α
2
(
n
)
α
3
(
n
)
⋯
⋯
α
n
(
n
)
]
[
1
0
0
⋯
⋯
⋯
0
0
0
−
1
1
0
⋯
⋯
⋯
0
0
0
0
−
1
1
⋯
⋯
⋯
0
0
0
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
0
0
0
0
⋯
⋯
−
1
1
0
0
0
0
0
0
⋯
0
−
1
1
]
−
1
[
λ
1
λ
2
−
λ
1
λ
3
−
λ
2
⋯
λ
n
−
1
−
λ
n
−
2
λ
n
−
λ
n
−
1
]
=
[
1
α
2
(
1
)
0
0
⋯
0
1
∑
r
=
2
3
α
r
(
2
)
α
3
(
2
)
0
⋯
0
⋯
⋯
⋯
⋯
⋯
⋯
1
∑
s
=
2
n
α
s
(
n
−
1
)
∑
u
=
3
n
α
u
(
n
−
1
)
⋯
⋯
α
n
(
n
−
1
)
1
∑
t
=
2
n
α
t
(
n
)
∑
w
=
3
n
α
w
(
n
)
⋯
⋯
α
n
(
n
)
]
[
λ
1
λ
2
−
λ
1
λ
3
−
λ
2
⋯
λ
n
−
1
−
λ
n
−
2
λ
n
−
λ
n
−
1
]
. From (4.4) we finally
obtain (4.5)
p
1
=
λ
1
+
(
λ
2
−
λ
1
)
α
2
(
1
)
,


p
2
=
λ
1
+
(
λ
2
−
λ
1
)
∑
r
=
2
3
α
r
(
2
)
+
(
λ
3
−
λ
2
)
α
3
(
2
)
,
…
,
p
n
=
λ
1
+
(
λ
2
−
λ
1
)
∑
t
=
2
n
α
t
(
n
)
+
(
λ
3
−
λ
2
)
∑
w
=
3
n
α
w
(
n
)
+
⋯
+
(
λ
n
−
λ
n
−
1
)
α
n
(
n
)
. Since “α”s and “∑”s in (4.5) are not certainly nonnegative and since
the sequence {
λ
m
}
(
m
=
1
,
…
,
n
) is increasing, then firstly by arbitrary selection of differences λ
2
−
λ
1
≥
0
,
λ
3
−
λ
2
≥
0
,
…
,
λ
n
−
1
−
λ
n
−
2
≥
0 and λ
n
−
λ
n
−
1
≥
0 and thereafter a sufficiently great λ
1
≥
0, all
hd margins p
m
(
m
=
1
,
…
,
n
) can be made nonnegative, that is, the matrix G
n can be always produced as truly hyperdominant. This completes the proof of this
theorem.Presentation of
explicit solution to the IEP of truly hd matrices with assigned nonnegative
spectrum is now in order. It has been proved in [16] that this IEP always has
at least one solution and that infinitely many others can be produced thereof
by using Givens rotations. Solution of that IEP is important in electrical
network synthesis of driving-point immittance functions and matrices of both passive
and active, common-ground, transformerless, two-element-kind R
L
C networks and in generation of various
classes of canonic and noncanonic equivalent realizations [19, 22]. In [16] we
have proved the existence of solution to the IEP of hd matrices with assigned nonnegative
spectrum, but here we shall present the explicit construction of solution matrix entries by using other
arguments. This represents the explicit solution of the problem opened in [17].Theorem 4.2.
For
any set of real nonnegative numbers{
λ
1
,
λ
2
,
…
,
λ
n
} there always exists at least one (and infinitely many) n
×
n real symmetric hyperdominant matrices
having these numbers as eigenvalues. In other words, IEP of symmetric hd
matrices with assigned nonnegative spectrum always has at least one solution.Proof.
We will take the same assumptions as
in Theorem4.1, except for θ
k
∈
(
0
,
π
/
2
)
(
k
=
1
,
…
,
n
−
1
). Through enumeration of eigenvalues, the nonnegative
sequence {
λ
m
}
(
m
=
1
,
…
,
n
) is made increasing. Then, according to Theorem 3.3 (Case 1), the symmetric
matrix G
n
=
U
G
1
U
T
[
G
1
=
diag
(
λ
1
,
λ
2
,
…
,
λ
n
)
] with spectrum {
λ
1
,
λ
2
,
…
,
λ
n
} and the entries
determined by (2.21), is produced with hd sign pattern, no matter what selection
of θ
k
s
(
k
=
1
,
…
,
n
−
1
) has been made. Observe that in Case 1
a
k
=
cos
⁡
θ
k and c
k
=
sin
⁡
θ
k. To make G
n
=
U
G
1
U
T truly hd, we will prove the existence of such
“θ”s that make all “α”s (and hence
all “p”s) in (4.1) nonnegative. Let we
introduce the following positive sequence {
M
m
}
(
m
=
1
,
…
,
n
)
(4.6)
M
1
:
=
a
1
+
a
2
c
1
+
a
3
c
2
c
1
+
⋯
+
a
n
−
1
c
n
−
2
⋯
c
2
c
1
+
c
n
−
1
c
n
−
2
⋯
c
2
c
1
M
2
:
=
a
2
+
a
3
c
2
+
a
4
c
3
c
3
+
⋯
+
a
n
−
1
c
n
−
2
⋯
c
3
c
2
+
c
n
−
1
c
n
−
2
⋯
c
3
c
2
M
3
:
=
a
3
+
a
4
c
3
+
a
5
c
4
c
3
+
⋯
+
a
n
−
1
c
n
−
2
⋯
c
4
c
3
+
c
n
−
1
c
n
−
2
⋯
c
4
c
3
M
n
−
2
:
=
a
n
−
2
+
a
n
−
1
c
n
−
2
+
c
n
−
1
c
n
−
2
,
M
n
−
1
:
=
a
n
−
1
+
c
n
−
1
,
M
n
:
=
1. Then, by
using (4.2) we obtain a consistent set of inequalities that ensure nonnegativity
of all “α”s in (4.1) (4.7)
α
1
(
1
)
=
a
1
M
1
≥
0
,


α
2
(
1
)
=
1
−
a
1
M
1
=
c
1
(
c
1
−
a
1
M
2
)
≥
0
,
(4.8)
α
1
(
k
)
≥
0
,


α
p
(
k
)
≥
0

−
c
p
−
1
+
a
p
−
1
M
p
≥
0

(
p
=
2
,
…
,
k
),
(4.9)
α
k
+
1
(
k
)
≥
0

c
k
−
a
k
M
k
+
1
≥
0

(
k
=
2
,
…
,
n
−
1
),
(4.10)
α
1
(
n
)
≥
0
,


α
q
(
n
)
≥
0

−
c
q
−
1
+
a
q
−
1
M
q
≥
0

(
q
=
2
,
…
,
n
−
1
),
(4.11)
α
n
(
n
)
≥
0

−
c
n
−
1
+
a
n
−
1
≥
0
,
(4.12)
α
n
(
n
−
1
)
=
b
n
−
1
2
+
b
n
−
1
d
n
−
1
=
c
n
−
1
2
−
a
n
−
1
c
n
−
1
=
c
n
−
1
(
c
n
−
1
−
a
n
−
1
)
≥
0

c
n
−
1
−
a
n
−
1
≥
0. For p
=
2 from (4.8) we obtain M
2
≥
c
1
/
a
1 and from (4.7) M
2
≤
c
1
/
a
1. Then, M
2
=
c
1
/
a
1
,
α
2
(
1
)
=
0
,
α
1
(
1
)
=
1
−
α
2
(
1
)
=
1
,
M
1
=
1
/
a
1 and p
1
=
λ
1. From (4.11)-(4.12) it follows that a
n
−
1
=
c
n
−
1
(
⇔
θ
n
−
1
=
π
/
4
) and α
n
(
n
−
1
)
=
α
n
(
n
)
=
0 [inequalty (4.12) is the same as (4.9) if k
=
n
−
1
(
M
n
=
1
)
]. For k
=
2
,
…
,
(
n
−
2
), we obtain from (4.9) M
k
+
1
≤
c
k
/
a
k and for q
=
3
,
…
,
(
n
−
1
), we obtain from (4.10) M
q
≥
c
q
−
1
/
a
q
−
1. To
summarize, we have proved that: (a) M
r
=
c
r
−
1
/
a
r
−
1, for r
=
2
,
…
,
(
n
−
1
) and (b) {
α
1
(
k
)
=
1
,
α
s
(
k
)
=
0
[
s
=
2
,
…
,
(
k
+
1
)
] and p
k
=
λ
1
}, for k
=
1
,
…
,
(
n
−
1
). And finally, from (4.10) we obtain α
q
(
n
)
=
0 for q
=
2
,
…
,
n
,
α
1
(
n
)
=
1 and p
n
=
λ
1. Since the
matrix G
n has hd sign pattern and each of its row (column) hd
margins is equal to λ
1
≥
0, then G
n is truly hd matrix. This completes the proof of the theorem.Remark 4.3.
It relates to calculation of entries ofG
n.
In Theorem 4.2 it is proved that M
1
=
1
/
a
1 and M
k
=
c
k
−
1
/
a
k
−
1
(
k
=
2
,
…
,
n
−
1
). It is assumed M
n
=
1.
Since θ
n
−
1
=
π
/
4, then M
n
−
1
=
a
n
−
1
+
c
n
−
1
=
cos
⁡
θ
n
−
1
+
sin
θ
n
−
1
=
2
⁡.
For w
=
1
,
…
,
n
−
1 it follows from (4.6) (4.13)
M
w
=
a
w
+
c
w
M
w
+
1
=
a
w
+
c
w
c
w
a
w
=
1
a
w
=
a
w
2
+
c
w
2
a
w
2
=
1
+
(
c
w
a
w
)
2
=
1
+
M
w
+
1
2
=
2
+
M
w
+
2
2
=
⋯
=
n
−
w
−
1
+
M
n
−
1
2
=
n
−
w
+
1
,
a
w
=
cos
θ
w
=
1
n
−
w
+
1
,


c
w
=
sin
θ
w
=
n
−
w
n
−
w
+
1
. By using (2.17)-(2.18),
(2.21), (4.13) we can easily calculate all entries of the (initial) hd matrix G
n.
Other hd matrices having the same spectrum can be produced thereof by application
of Givens rotations, one at a time.


## 5. Application of the Obtained Results in Electrical Network Synthesis

It is well known that synthesis
methods of passive, common-ground, transformerless, two-element-kindR
L
C networks yield topological configurations which are severely restricted
by the method chosen [19]. By using  of
the results above, a new class of non-canonic, driving-point immittance
realizations of passive, common-ground, transformerless, two-element-kind R
L
C networks with minimum number of both
nodes and elements of one kind can be generated with possibility of reduction
in number of elements of other kind. The network synthesis is always performed
by using normalization of both the complex frequency s and the impedance Z
(
s
).
If Ω is a selected normalization frequency, then
the normalized frequency is s
n
=
s
/
Ω.
Similarly, if R
0 is a selected normalization resistance, then
the normalized impedance is Z
n
(
s
)
=
Z
(
s
)
/
R
0.
Thereby we achieve [20]: (a) lesser dispersion of coefficients in normalized
functions and (b) dimensionless manipulation of quantities. The normalized
resistance of resistor R is R
n
:
=
R
/
R
0.
The normalized impedance of an inductor L is Z
L
n
(
s
)
=
L
s
/
R
0
=
(
L
Ω
/
R
0
)
s
n
=
L
n
s
n (L
n
:
=
L
Ω
/
R
0-normalized inductance). The normalized
impedance of a capacitor C is Z
C
n
(
s
)
=
1
/
(
C
R
0
s
)
=
1
/
[
(
R
0
C
Ω
)
s
n
]
=
1
/
(
C
n
s
n
) (C
n
:
=
R
0
C
Ω-normalized capacitance). To physically
realize a network after synthesis, a denormalization process must be performed. The actual parameter values of R
L
C elements are calculated as follows: R
=
R
n
R
0
,
L
=
L
n
R
0
/
Ω
,
C
=
C
n
/
(
R
0
Ω
). From now on it will be assumed that
normalized synthesis is being carried out, but the lower index “n’’ we be dropped from component labels for
brevity.It is well known that if a real
rational function ins can be realized as R
L driving-point impedance Z
R
L
(
s
),
then it can be also realized as R
C driving-point admittance Y
R
C
(
s
) [20]. And similarly, if it can be realized as R
L driving-point admittance Y
R
L
(
s
),
then it can also be realized as R
C driving-point
impedance Z
R
C
(
s
).
The L
C
:
R
C transformation turns the synthesis of L
C driving-point impedance  Z
L
C
□
(
s
) to synthesis of R
C driving-point impedance  Z
R
C
(
s
)
=
Z
L
C
□
(
s
)
/
s [20]. It also turns the synthesis of L
C driving-point admittance  Y
L
C
□
(
s
) to synthesis of R
C driving-point admittance  Y
R
C
(
s
)
=
s
⁢
⋅
Y
L
C
□
(
s
)
. These R
C driving-point imittances are at first
realized by prototype R
C networks and
thereof are produced the desired L
C networks in the following way: capacitors in R
C and
L
C networks remain
the same, but the resistor from R
C network turns to inductor in L
C network with the same parameter value. Also, L
R
:
R
C transformation
turns the synthesis of R
L driving-point impedance Z
R
L
□
(
s
) to synthesis of R
C driving-point impedance  Z
R
C
(
s
)
=
Z
R
L
□
(
s
)
/
s
. It also turns synthesis of R
L driving-point admittance Y
R
L
□
(
s
) to synthesis of R
C driving-point admittance Y
R
C
(
s
)
=
s
Y
R
L
□
(
s
)
. These R
C imittances are realized by prototype R
C networks and the desired R
L networks
are produced thereof in the following way: the resistor from R
C network turns to inductor in L
R network with the same parameter
value, and the capacitor from R
C network turns to resistor in L
R network
with reciprocal parameter value. Bearing all the aforementioned on mind, we can
obviously restrict our consideration only to synthesis of driving-point impedance functions Z
R
C
(
s
) of R
C networks,
which satisfy the following well known analytic necessary and sufficient
conditions [20]: (a) Z
R
C
(
s
) is real rational function in s, (b) It has only simple poles on negative real axis, or at the origin. At infinity
it cannot have pole and (c) Residues
of these poles are real and positive and A
∞
:
=
lim
⁡
s
→
∞
⁢
Z
R
C
(
s
)
≥
0.In general, the first canonic
Foster's expansion (form) ofZ
R
C
(
s
) [20] reads(5.1)
Z
R
C
(
s
)
=
A
∞
+
∑
m
=
0
n
A
m
s
+
s
m

[
A
∞
>
0
;
s
0
=
0
,
A
0
>
0
;
s
p
,
A
p
>
0
⁢
(
p
=
1
,
…
,
n
)
]
, where A
m is residue of the pole s
m
(
m
=
0
,
1
,
…
,
n
).
The network which realizes driving-point impedance Z
R
C
(
s
) (5.1) with minimum number of nodes (
=
n
+
1
),
minimum number of resistors (
=
n
+
1
) and minimum number of capacitors (
=
n
+
1
) is depicted in Figure 1. Observe that neither
the resistors, nor the capacitors share common-node and hence the overall
network realization is said to be non common-grounded.Figure 1
The first canonic
Foster's realization ofZ
R
C
(
s
). Denoted
are the normalized “values” of R
C parameters.Now, we will present our synthesis
procedure. If for a given driving-point impedanceZ
R
C
(
s
) we found that A
0
>
0 and/or A
∞
>
0,
then in the preamble of the realization procedure A
∞ and/or A
0
/
s should be at first extracted from (5.1) and
realized by a series connection of resistor R
∞
=
A
∞ and capacitor C
0
=
1
/
A
0,
thereby leaving for realization the driving-point impedance Z
¯
R
C
(
s
)
=
Z
R
C
(
s
)
−
A
∞
−
A
0
/
s with solely n poles lying
on the negative real axis. In the sequel we will assume that Z
R
C
(
s
) has only n such poles.LetC
=
diag
(
C
1
,
C
2
,
…
,
C
n
) and G
=
diag
(
G
1
,
G
2
,
…
,
G
n
) be diagonal n
×
n matrices with strictly positive diagonal
entries corresponding to the normalized capacitances and conductances,
respectively. If we arbitrarily choose a nonsingular n
×
n matrix T,
then the reciprocal passive networks which realize C
s
+
G and Y
(
s
)
=
T
(
C
s
+
G
)
T
T will have the same natural frequencies. By
arbitrary selection of n
×
n nonsingular diagonal matrices δ
=
diag
(
δ
1
,
δ
2
,
…
,
δ
n
),
a broad class of nonsingular n
×
n matrices T can be generated with assumption T
=
V
δ
U,
where U and V are n
×
n orthogonal matrices. Since Y
(
s
)
=
T
(
C
s
+
G
)
T
T
=
V
[
δ
(
U
C
U
T
)
δ
s
+
δ
(
U
G
U
T
)
δ
]
V
T,
then(5.2)
Z
(
s
)
=
[
z
m
r
(
s
)
]
n
×
n
=
Y
−
1
(
s
)
=
[
T
(
C
s
+
G
)
T
T
]
−
1
=
(
V
δ
−
1
U
C
1
/
2
)
(
s
I
n
+
G
C
−
1
)
−
1
(
V
δ
−
1
U
C
1
/
2
)
T
.Various network topologies can be
produced by different choices of U and V.
But, only by selecting C
=
C
I
n
⁢
(
C
>
0
) and V
=
I
n,
the networks with minimum number of common-ground capacitors are produced; and
only by selecting G
=
G
I
n
⁢
(
G
>
0
) and V
=
I
n,
the networks with minimum number of common-ground resitors are produced. Let us select C
=
C
I
n
⁢
(
C
>
0
) and V
=
I
n,
and let us assume in (5.1): A
0
=
A
∞
=
0 and s
n
>
s
n
−
1
⋯
>
s
1
>
0. Since U
=
[
u
m
r
]
(
m
,
r
=
1
,
…
,
n
),
then from (5.2) it follows that(5.3)
Y
(
s
)
=
C
δ
2
s
+
δ
(
U
G
U
T
)
δ
,


z
m
r
(
s
)
=
1
C
δ
m
δ
r
∑
p
=
1
n
u
m
p
u
r
p
s
+
G
p
/
C

(
m
,
r
=
1
,
…
,
n
)
. The matrices which are effectively
realized by common-ground network with n
+
1 nodes [(
n
+
1
)th node is the common-ground] are C
δ
2
s and
δ
(
U
G
U
T
)
δ
, provided that both are truly hd. According to
(5.1) and (5.3) it holds G
p
=
C
s
p
⁢
(
p
=
1
,
…
,
n
) and G
n
>
G
n
−
1
⋯
>
G
1
>
0. By using Theorem 3.3
[Case 1, P
k
=
A
k and θ
k
∈
(
0
,
π
/
2
)
(
k
=
1
,
…
,
n
−
1
)] we infer that U
G
U
T (2.21) is produced with hd sign pattern and no
zero entries and with strictly positive inverse. Matrix U (2.26) is lower Hessenberg with nonnegative
entries, except for negative “b”s.
The same conclusions relating to U
G
U
T and U also hold if we apply Case 1 of Theorem 3.3 with P
k
=
B
k and θ
k
∈
(
0
,
π
/
2
)
(
k
=
1
,
…
,
n
−
1
),
except for “d”s in (2.26) are then
negative and “b”s are positive. To realize Z
R
C
(
s
) we must select in (5.3) either m
=
r
=
n
−
1 or m
=
r
=
n
, thus obtaining either Z
R
C
(
s
)
=
z
n
−
1
,
n
−
1
(
s
)
, or Z
R
C
(
s
)
=
z
n
n
(
s
)
. By assuming m
=
r
=
n
[
⇔
Z
R
C
(
s
)
=
z
n
n
(
s
)
]
,
P
k
=
A
k and θ
k
∈
(
0
,
π
/
2
)
(
k
=
1
,
…
,
n
−
1
)
, it
follows from (2.26),
(5.1), (5.3)(5.4)
C
δ
n
2
=
1
∑
p
=
1
n
A
p
,


a
i
−
1
=
d
i
−
1
=
(
A
i
∑
q
=
1
i
A
q
)
1
/
2
,

c
i
−
1
=
(
∑
r
=
1
i
−
1
A
r
∑
q
=
1
i
A
q
)
1
/
2
,

(
i
=
2
,
…
,
n
)
. To prove the existence of a physical
realization of both C
δ
2
s and δ
(
U
G
U
T
)
δ we still have to determine the positive
column  vector col
(
δ
)
=
[
δ
1
δ
2
⋯
δ
n
]
T
δ
1
δ
2
⋯
δ
n
]
T which, according to
Theorem 3.5, makes δ
(
U
G
U
T
)
δ truly hd with possibly zero hd margins of at
most n
−
1 rows. Let hd margin of the ith row in δ
(
U
G
U
T
)
δ be p
i
(
i
=
1
,
…
,
n
) and let p
:=
[
p
1
p
2
⋯
p
n
]
T
p
1
p
2
⋯
p
n
]
T.
If u
:=
[
1
1
⋯
1
]
T
1
1
⋯
1
]
T (n unities), then p
=
δ
(
U
G
U
T
)
δ
u
. Let we introduce a column vector p
∗
:=
[
p
1
∗
p
2
∗
⋯
p
n
∗
]
T
p
1
∗
p
2
∗
⋯
p
n
∗
]
T  (∥
p
∗
∥
E
>
0) of arbitrarily
assumed real nonegative numbers. Since (
U
G
U
T
)
−
1 is strictly positive, then it always can be
find a diagonal matrix δ
=
diag
(
δ
1
,
δ
2
,
…
,
δ
n
) with positive diagonal entries, such that p
*
=
(
U
G
U
T
)
δ
u
=
(
U
G
U
T
)
col
(
δ
).
Herefrom, we obtain col
(
δ
)
=
(
U
G
U
T
)
−
1
p
*, that is, that δ
i
>
0
⁢
(
i
=
1
,
…
,
n
). Since
p
=
δ
(
U
G
U
T
)
δ
u
=
δ
p
∗,
then it follows p
i
=
δ
i
p
i
∗
≥
0
⁢
(
i
=
1
,
…
,
n
),   bearing on mind that at most n
−
1 “p”s
can be equal to zero. These “p”s indices correspond to indices of those rows
(or columns) in δ
(
U
G
U
T
)
δ which have zero hd margins. Then, from the
overall network vanish resistors connecting common-ground to nodes with the
same indices as that of rows (columns) with zero hd margins [22]. For different
selections of p
*,
different algorithms and different topologically and parametrically equivalent
realizations emerge. For example, if we
select col
(
δ
)
=
μ
[
u
11
u
21
⋯
u
n
1
]
T
u
11
u
21
⋯
u
n
1
]
T
⁢
(
μ
>
0
)
, where it is according to (2.26),
(5.4)(5.5)
u
11

=


[
A
2
A
1

+

A
2

]
1
/
2
,



u
p
1

=


[

A
1
A
p
+
1
(
∑
q
=
1
n
A
q
)
(
∑
r
=
1
p
+
1
A
r
)
]
1
/
2


(
p

=

2
,
…
,
n

−

1
)
,



u
n
1

=


[

A
1
∑
q
=
1
n
A
q

]
1
/
2
, then the column vector p of row (column) hd margins of matrix δ
(
U
G
U
T
)
δ with hd sign pattern reads(5.6)
p
=
δ
(
U
G
U
T
)
δ
u
=
δ
(
U
G
U
T
)col(
δ
)
=
G
1
col(
δ
2
)
>
0
n
,
1
. This means that δ
(
U
G
U
T
)
δ is truly hd. We will now present two
algorithms for realization of driving-point impedances Z
R
C
(
s
) which rely on the results developed above.Algorithm 1.
Realization ofZ
R
C
(
s
)
with
minimum number of common-ground capacitors and non-reduced number of
resistors(10) Commencing with A
i
⁢
(
i
=
1
,
…
,
n
) calculate the entries of U,
by using (2.26) and (5.4).(20) Arbitrarily select
some μ
>
0 and then calculate C
=
1
/
μ
2
A
1 and G
q
=
C
s
q
⁢
(
q
=
1
,
…
,
n
).(30) Calculate col
(
δ
)
=
μ
[
u
11
u
21
⋯
u
n
1
]
T
u
11
u
21
⋯
u
n
1
]
T
, by using (5.5) and the entries of hd matrix δ
(
U
G
U
T
)
δ
. Calculate p,
by using (5.6).(40) Realize δ
(
U
G
U
T
)
δ by common-ground, transformerless, conductance
network. This can be done easily, almost by visual inspection of δ
(
U
G
U
T
)
δ [22]. Attach to the ports of that network,
enumerated by 1, p
(
p
=
2
,
…
,
n
−
1
) and n,
the common-ground capacitors with normalized capacitances(5.7)
C
1
=
A
2
A
1
(
A
1
+
A
2
)
,


C
p
=
A
p
+
1
(
∑
q
=
1
n
A
q
)
(
∑
r
=
1
p
+
1
A
r
)

(
p
=
2
,
…
,
n
−
1
)
,


C
n
=
1
∑
q
=
1
n
A
q
, respectively. The nth  port
of the overall network realizes driving-point impedance Z
R
C
(
s
),
provided that all other ports are left open-circuited.Algorithm 2.
Realization ofZ
R
C
(
s
)
with
minimum number of common-ground capacitors and the reduced number of
resistors
(10) The same as step (
1
0
) of Algorithm 1. Let S
=
diag
(
s
1
,
s
2
,
…
,
s
n
).
Recall that s
n
>
s
n
−
1
⋯
>
s
1
>
0.
(20) Select ε
1
>
0 and ε
i
=
0
(
i
=
2
,
…
,
n
−
1
)
. Thereafter, by using (2.21) and (5.4), calculate (5.8)
C
=
[
ε
1
(
s
1
−
1
−
s
2
−
1
)
c
n
−
1
c
n
−
2
⋯
c
2
c
1
a
1
]
2
∑
q
=
1
n
A
q
,


G
i
=
C
s
i

(
i
=
1
,
…
,
n
)
.Calculate col
(
δ
)
=
ε
1
C
−
1
(
U
S
−
1
U
T
)
e
1,
where e
1
=
[
1
0
⋯
0
]
T
1
0
⋯
0
]
T is n-dimensional column vector.
(30) Calculate the entries
of δ
(
U
G
U
T
)
δ and its hd margin p
1
=
δ
1
ε
1
, where δ
1
=
ε
1
C
−
1
e
1
T
T
S
−
1
T
T
e
1
. Set for other hd margins p
i
=
0
⁢
(
i
=
2
,
…
,
n
)
.
(40) Realize δ
(
U
G
U
T
)
δ by common-ground, transformerless, conductance
network and attach to its ith port
the common-ground capacitor with normalized capacitance C
δ
i
2
⁢
(
i
=
1
,
…
,
n
)
. The nth
port of the overall network realizes driving-point impedance Z
R
C
(
s
)
, provided that all other ports are left
open-circuited.


### 5.1. A Numerical Example

Consider realization of the real
rational functionZ
L
C
□
(
s
) as driving-point impedance of common-ground
transformerless
L
C network with minimum number of
capacitors and the reduced number of inductors(5.9)
Z
L
C
□
(
s
)
=
(
s
2
+
1
)
(
s
2
+
3
)
(
s
2
+
5
)
(
s
2
+
8
)
s
(
s
2
+
2
)
(
s
2
+
4
)
(
s
2
+
6
)
.This function satisfies the necessary
and sufficient conditions for driving-point immittances of L
C networks: (a) it is an odd real rational function in s;
(b) it has only simple poles located strictly on imaginary axis; and (c)
residues of those poles are real and positive. Therefore, Z
L
C
□
(
s
) can be realized both in two Foster's and in
two Cauer's canonic forms [20]. The partial fraction expansion of Z
L
C
□
(
s
) reads(5.10)
Z
L
C
□
(
s
)
=
(
s
2
+
1
)
(
s
2
+
3
)
(
s
2
+
5
)
(
s
2
+
8
)
s
(
s
2
+
2
)
(
s
2
+
4
)
(
s
2
+
6
)
=
s
+
5
2
s
+
9
s
/
8
s
2
+
2
+
3
s
/
4
s
2
+
4
+
5
s
/
8
s
2
+
6
. The reactance function corresponding
to Z
L
C
□
(
s
) is X
L
C
□
(
ω
)
:
=
Z
L
C
□
(
j
ω
)
/
j and it is depicted in Figure 2. The first
canonic Foster's realization of Z
L
C
□
(
s
) with minimum number of nodes, noncommon-ground
capacitors and inductors is depicted in Figure 3. Thereon are denoted the
normalized values of L
C parameters.Figure 2
The
reactance function ofL
C network from the example.Figure 3
Driving-point impedanceZ
L
C
□
(
s
) from example synthesized by using the first
canonic Foster's form.In Figure4, it is depicted the first
canonic Foster's realization of driving-point impedance Z
L
C
□
(
s
) from Figure 3 with selected normalization
frequency Ω
=
1
0
6
[
rad
/
s
] and selected normalization resistance R
0
=
1
0
3
[
k
Ω
].
The network is excited by a sinusoidal current generator having constant
current amplitude and discretely varying frequency f
=
ω
/
(
2
π
) within the range f
∈
[
0.1
,
5
00
] kHz. If the complex representative of
generator current is I
̲
g and the complex representative of the voltage
across its terminals is U
̲
g, then the complex driving-point impedance of
the overall L
C network is Z
¯
L
C
□
=
U
¯
g
/
I
¯
g
. The modulus of Z
¯
L
C
□
, that is, |
Z
¯
L
C
□
| (usually called L
C impedance) obtained through PSPICE simulation
within the range f
∈
[
0.
1
,
5
00
] kHz is depicted in Figure 5.Figure 4
Denormalized realization of driving-point impedanceZ
L
C
□
(
s
) from Figure
3.Figure 5
Driving-point impedance|
Z
¯
L
C
□
| of L
C network from Figure  4.Now, we will realizeZ
L
C
□
(
s
) by using the proposed
Algorithm 2. After L
C
:
R
C transformation, we firstly produce the
function Z
¯
R
C
(
s
)
=
Z
L
C
□
(
s
)
/
s
=
1
+
5
/
2
s
+
Z
R
C
(
s
)
, where Z
R
C
(
s
) is driving-point impedance of R
C network which should be expanded into
partial fractions as follows:(6.1)
Z
R
C
(
s
)
=
A
1
s
+
s
1
+
A
2
s
+
s
2
+
A
3
s
+
s
3
|
A
1
=
9
8
,

A
2
=
3
4
,

A
3
=
5
8
,
s
1
=
2
,

s
2
=
4
,

s
3
=
6
,
s
3
>
s
2
>
s
1
>
0
,

S
:
=
diag
(
s
1
,
s
2
,
s
3
)
. In step (
1
0
) of Algorithm 2 we determine the orthogonal matrix U by using A
1
,
A
2
, and A
3 (see (2.26) and (5.4))(6.2)
U
=
[
2
5
−
3
5
0
3
20
1
10
−
3
2
9
20
3
10
1
2
]
.By assuming ε
1
=
8
5
/
3 in step (
2
0
),
we further easily obtain C
=
1
,
G
1
=
2
,
G
2
=
4
,
G
3
=
6 and col
(
δ
)
=
[
δ
1
δ
2
δ
3
]
T
δ
1
δ
2
δ
3
]
T
=
[
2
.0
87
0.
365
0.
632
]
T
2
.0
87
0.
365
0.
632
]
T
. In step (
3
0
) we firstly calculate δ
(
U
G
U
T
)
δ and then hd margins of its rows (columns),(6.3)
δ
(
U
G
U
T
)
δ
=
[
13
.937
−
0.373
−
1.119
−
0.373
0.693
−
0.320
−
1.119
−
0.320
1.439
]
,


p
1
=
12
.444
,


p
2
=
p
3
=
0. In step (
4
0
) we calculate the normalized capacitances
of common-ground capacitors: C
1
=
4
.
355
,
C
2
=
0.
133 and C
3
=
0.
4
00. Realization of driving-point impedance Z
R
C
(
s
) by transformerless, common-ground R
C network with minimum number of nodes (
=
n
+
1
)
, reduced number of inductors and minimum number
of common-ground capacitors (
=
n
)
, begins by realization of conductance matrix δ
(
U
G
U
T
)
δ which can be accomplished almost by inspection
of that matrix [22]. Then, to the ith port of the realized conductance network,
it should be connected to the capacitor C
i
=
δ
i
2
⁢
(
i
=
1
,
2
,
3
)
. The third port of the overall network
realizes the R
C driving-point impedance Z
R
C
(
s
)
, provided that all other ports are left
open-circuited. By embedding in the third port a series connection of resistor
and capacitor with the normalized parameter values 1 and 2/5, respectively, and
by applying R
C
:
L
C transformation thereafter, we finally produce
noncanonic network which realizes Z
L
C
□
(
s
) with minimum number of nodes and capacitors
and with reduced number of inductors. That network is depicted in Figure 6
whereon are denoted the normalized (dimensionless) values of L
C parameters.Figure 6
Driving-point impedanceZ
L
C
□
(
s
) from the example synthesized by a noncanonic L
C network.In Figure7,
is depicted the noncanonic realization of driving-point impedance Z
L
C
□
(
s
) from Figure 6 with selected normalization
frequency Ω
=
1
0
6
[
rad
/
s
] and selected normalization resistance R
0
=
1
0
3
[
k
Ω
]
. The network is excited by a sinusoidal current
generator with constant current amplitude and with discretely variable
frequency f
=
ω
/
(
2
π
) within the range f
∈
[
0.
1
,
5
00
] kHz. If the complex representative of
generator current is I
¯
g and the complex representative of the voltage
across its terminals is U
¯
g,
then the complex driving-point impedance of the overall L
C network is Z
¯
L
C
□
=
U
¯
g
/
I
¯
g
. The modulus of Z
¯
L
C
□
, that is, |
Z
¯
L
C
□
| (usually called L
C impedance) obtained through PSPICE simulation
within the range f
∈
[
0.
1
,
5
00
] kHz is depicted in Figure 8. Since L
C networks
in Figures 4 and
7 are intentionally designed to be equivalent, then their
driving-point impedances |
Z
¯
L
C
□
| must have the same variations in frequency, as
can be verified from Figures 5 and
8 qualitatively and more precisely by using
numerical results of simulation.Figure 7
Denormalized realization of driving-point impedanceZ
L
C
□
(
s
) from Figure 6.Figure 8
Driving-point impedance|
Z
¯
L
C
□
| of L
C network from Figure 7.


## 5.1. A Numerical Example

Consider realization of the real
rational functionZ
L
C
□
(
s
) as driving-point impedance of common-ground
transformerless
L
C network with minimum number of
capacitors and the reduced number of inductors(5.9)
Z
L
C
□
(
s
)
=
(
s
2
+
1
)
(
s
2
+
3
)
(
s
2
+
5
)
(
s
2
+
8
)
s
(
s
2
+
2
)
(
s
2
+
4
)
(
s
2
+
6
)
.This function satisfies the necessary
and sufficient conditions for driving-point immittances of L
C networks: (a) it is an odd real rational function in s;
(b) it has only simple poles located strictly on imaginary axis; and (c)
residues of those poles are real and positive. Therefore, Z
L
C
□
(
s
) can be realized both in two Foster's and in
two Cauer's canonic forms [20]. The partial fraction expansion of Z
L
C
□
(
s
) reads(5.10)
Z
L
C
□
(
s
)
=
(
s
2
+
1
)
(
s
2
+
3
)
(
s
2
+
5
)
(
s
2
+
8
)
s
(
s
2
+
2
)
(
s
2
+
4
)
(
s
2
+
6
)
=
s
+
5
2
s
+
9
s
/
8
s
2
+
2
+
3
s
/
4
s
2
+
4
+
5
s
/
8
s
2
+
6
. The reactance function corresponding
to Z
L
C
□
(
s
) is X
L
C
□
(
ω
)
:
=
Z
L
C
□
(
j
ω
)
/
j and it is depicted in Figure 2. The first
canonic Foster's realization of Z
L
C
□
(
s
) with minimum number of nodes, noncommon-ground
capacitors and inductors is depicted in Figure 3. Thereon are denoted the
normalized values of L
C parameters.Figure 2
The
reactance function ofL
C network from the example.Figure 3
Driving-point impedanceZ
L
C
□
(
s
) from example synthesized by using the first
canonic Foster's form.In Figure4, it is depicted the first
canonic Foster's realization of driving-point impedance Z
L
C
□
(
s
) from Figure 3 with selected normalization
frequency Ω
=
1
0
6
[
rad
/
s
] and selected normalization resistance R
0
=
1
0
3
[
k
Ω
].
The network is excited by a sinusoidal current generator having constant
current amplitude and discretely varying frequency f
=
ω
/
(
2
π
) within the range f
∈
[
0.1
,
5
00
] kHz. If the complex representative of
generator current is I
̲
g and the complex representative of the voltage
across its terminals is U
̲
g, then the complex driving-point impedance of
the overall L
C network is Z
¯
L
C
□
=
U
¯
g
/
I
¯
g
. The modulus of Z
¯
L
C
□
, that is, |
Z
¯
L
C
□
| (usually called L
C impedance) obtained through PSPICE simulation
within the range f
∈
[
0.
1
,
5
00
] kHz is depicted in Figure 5.Figure 4
Denormalized realization of driving-point impedanceZ
L
C
□
(
s
) from Figure
3.Figure 5
Driving-point impedance|
Z
¯
L
C
□
| of L
C network from Figure  4.Now, we will realizeZ
L
C
□
(
s
) by using the proposed
Algorithm 2. After L
C
:
R
C transformation, we firstly produce the
function Z
¯
R
C
(
s
)
=
Z
L
C
□
(
s
)
/
s
=
1
+
5
/
2
s
+
Z
R
C
(
s
)
, where Z
R
C
(
s
) is driving-point impedance of R
C network which should be expanded into
partial fractions as follows:(6.1)
Z
R
C
(
s
)
=
A
1
s
+
s
1
+
A
2
s
+
s
2
+
A
3
s
+
s
3
|
A
1
=
9
8
,

A
2
=
3
4
,

A
3
=
5
8
,
s
1
=
2
,

s
2
=
4
,

s
3
=
6
,
s
3
>
s
2
>
s
1
>
0
,

S
:
=
diag
(
s
1
,
s
2
,
s
3
)
. In step (
1
0
) of Algorithm 2 we determine the orthogonal matrix U by using A
1
,
A
2
, and A
3 (see (2.26) and (5.4))(6.2)
U
=
[
2
5
−
3
5
0
3
20
1
10
−
3
2
9
20
3
10
1
2
]
.By assuming ε
1
=
8
5
/
3 in step (
2
0
),
we further easily obtain C
=
1
,
G
1
=
2
,
G
2
=
4
,
G
3
=
6 and col
(
δ
)
=
[
δ
1
δ
2
δ
3
]
T
δ
1
δ
2
δ
3
]
T
=
[
2
.0
87
0.
365
0.
632
]
T
2
.0
87
0.
365
0.
632
]
T
. In step (
3
0
) we firstly calculate δ
(
U
G
U
T
)
δ and then hd margins of its rows (columns),(6.3)
δ
(
U
G
U
T
)
δ
=
[
13
.937
−
0.373
−
1.119
−
0.373
0.693
−
0.320
−
1.119
−
0.320
1.439
]
,


p
1
=
12
.444
,


p
2
=
p
3
=
0. In step (
4
0
) we calculate the normalized capacitances
of common-ground capacitors: C
1
=
4
.
355
,
C
2
=
0.
133 and C
3
=
0.
4
00. Realization of driving-point impedance Z
R
C
(
s
) by transformerless, common-ground R
C network with minimum number of nodes (
=
n
+
1
)
, reduced number of inductors and minimum number
of common-ground capacitors (
=
n
)
, begins by realization of conductance matrix δ
(
U
G
U
T
)
δ which can be accomplished almost by inspection
of that matrix [22]. Then, to the ith port of the realized conductance network,
it should be connected to the capacitor C
i
=
δ
i
2
⁢
(
i
=
1
,
2
,
3
)
. The third port of the overall network
realizes the R
C driving-point impedance Z
R
C
(
s
)
, provided that all other ports are left
open-circuited. By embedding in the third port a series connection of resistor
and capacitor with the normalized parameter values 1 and 2/5, respectively, and
by applying R
C
:
L
C transformation thereafter, we finally produce
noncanonic network which realizes Z
L
C
□
(
s
) with minimum number of nodes and capacitors
and with reduced number of inductors. That network is depicted in Figure 6
whereon are denoted the normalized (dimensionless) values of L
C parameters.Figure 6
Driving-point impedanceZ
L
C
□
(
s
) from the example synthesized by a noncanonic L
C network.In Figure7,
is depicted the noncanonic realization of driving-point impedance Z
L
C
□
(
s
) from Figure 6 with selected normalization
frequency Ω
=
1
0
6
[
rad
/
s
] and selected normalization resistance R
0
=
1
0
3
[
k
Ω
]
. The network is excited by a sinusoidal current
generator with constant current amplitude and with discretely variable
frequency f
=
ω
/
(
2
π
) within the range f
∈
[
0.
1
,
5
00
] kHz. If the complex representative of
generator current is I
¯
g and the complex representative of the voltage
across its terminals is U
¯
g,
then the complex driving-point impedance of the overall L
C network is Z
¯
L
C
□
=
U
¯
g
/
I
¯
g
. The modulus of Z
¯
L
C
□
, that is, |
Z
¯
L
C
□
| (usually called L
C impedance) obtained through PSPICE simulation
within the range f
∈
[
0.
1
,
5
00
] kHz is depicted in Figure 8. Since L
C networks
in Figures 4 and
7 are intentionally designed to be equivalent, then their
driving-point impedances |
Z
¯
L
C
□
| must have the same variations in frequency, as
can be verified from Figures 5 and
8 qualitatively and more precisely by using
numerical results of simulation.Figure 7
Denormalized realization of driving-point impedanceZ
L
C
□
(
s
) from Figure 6.Figure 8
Driving-point impedance|
Z
¯
L
C
□
| of L
C network from Figure 7.


## 6. Conclusions

A novel procedure for explicit
construction of entries of real symmetric matrices with assigned spectrum is developed
by using a group of four types of canonic, second-order, orthogonal
transformations. It has been also shown that the orthogonal modal matrices corresponding
to the produced real symmetrix matrices, are either lower or upper Hessenberg
with explicitly constructed entries too. Thereafter, the inverse eigenvalue
problems of real symmetric matrices with twelve specific types of sign patterns
(including hyperdominant one) are explicitly solved providing that the signs of
eigenvalues are the same (zeros are permitted) and that they are enumerated
such as to establish the increasing or decreasing sequence. It is proved to
arise thereof a possibility of explicit solving the inverse eigenvalue problem
of symmetric hyperdominant matrices having either uncommitted or assigned nonnegative
spectrum. The results obtained are then applied in synthesis of driving-point immittance functions of
transformerless, common-ground, two-element-kindR
L
C networks and in generation of their equivalent
realizations with minimum number of nodes. The synthesis procedures proposed
herein turn the synthesis problem of any immittance function of the
two-element-kind R
L
C network to the synthesis problem of impedance
function of a prototype R
C network.



---

*Source: 513582-2008-09-28.xml*