Brauer’s ovals theorem

Brauer’s ovals theorem

Let

A

be a square

complex matrix

,

R

i

=

∑

j

≠

i

|

a

i

⁢

j

|

1

≤

i

≤

n

. Let’s consider the ovals of this kind:

O

i

⁢

j

=

{

z

∈

ℂ

:

|

z

-

a

i

⁢

i

|

⁢

|

z

-

a

j

⁢

j

|

≤

R

i

⁢

R

j

}

∀

i

≠

j

. Such ovals are called

Cassini ovals

.

Theorem

(A. Brauer): All the

eigenvalues

of A lie inside the union of these

n

⁢

(

n

-

1

)

2

ovals of Cassini:

σ

⁢

(

A

)

⊆

⋃

i

≠

j

O

i

⁢

j

.

Proof:
Let

(

λ

,

𝐯

)

be an eigenvalue-eigenvector pair for

A

, and let

v

p

,

v

q

be the

components

of

𝐯

with the two maximal

absolute values

, that is

|

v

p

|

≥

|

v

q

|

≥

|

v

i

|

∀

i

≠

p

. (Note that

|

v

p

|

≠

0

, otherwise

𝐯

should be all-zero, in contrast with

eigenvector

definition

). We can also assume that

|

v

q

|

is not zero, because otherwise

A

⁢

𝐯

=

λ

⁢

𝐯

would imply

a

p

⁢

p

=

λ

, which trivially verifies the thesis. Then, since

A

⁢

𝐯

=

λ

⁢

𝐯

, we have:

(

λ

-

a

p

⁢

p

)

⁢

v

p

=

∑

j

=

1

,

j

≠

p

n

a

p

⁢

j

⁢

v

j

and so

|

λ

-

a

p

⁢

p

|

⁢

|

v

p

|

=

|

∑

j

=

1

,

j

≠

p

n

a

p

⁢

j

⁢

v

j

|

≤

∑

j

=

1

,

j

≠

p

n

|

a

p

⁢

j

|

⁢

|

v

j

|

≤

∑

j

=

1

,

j

≠

p

n

|

a

p

⁢

j

|

⁢

|

v

q

|

=

R

p

⁢

|

v

q

|

that is

|

λ

-

a

p

⁢

p

|

≤

R

p

⁢

|

v

q

|

|

v

p

|

.

In the same way, we obtain:

|

λ

-

a

q

⁢

q

|

≤

R

q

⁢

|

v

p

|

|

v

q

|

.

Multiplying the two

inequalities

, the two fractional terms vanish, and we get:

|

λ

-

a

p

⁢

p

|

⁢

|

λ

-

a

q

⁢

q

|

≤

R

p

⁢

R

q

which is the thesis.

□

Remarks:

1) Much like the

Levy-Desplanques theorem

states a

sufficient

condition, based on Gerschgorin circles, for non-singularity of a matrix, Brauer’s theorem can be employed to state a

similar

sufficient condition; namely, the following result of Ostrowski holds:

Corollary: Let

A

be a

n

×

n

complex-valued matrix; if for all

i

≠

j

we have

|

a

i

⁢

i

|

⁢

|

a

j

⁢

j

|

>

R

i

⁢

R

j

, then

A

is non

singular

.

The proof is

obvious

, since, by Brauer’s theorem, the above condition excludes the point

z

=

0

from the

spectrum

of

A

, implying this way

det

⁡

(

A

)

≠

0

.

2) Since both Gerschgorin’s and Brauer’s results rely upon the same

2

⁢

n

numbers, namely

{

a

i

⁢

i

}

i

=

1

n

and

{

R

i

}

i

=

1

n

, one may wonder if Brauer’s result is stronger than Gerschgorin’s one; actually, the answer is

positive

, as the following

inclusion

shows:

Corollary: Let

G

⁢

(

A

)

=

⋃

i

=

1

n

D

i

⁢

(

A

)

and

B

⁢

(

A

)

=

⋃

i

≠

j

n

O

i

⁢

j

⁢

(

A

)

be respectively Gershgorin and Brauer eigenvalues inclusion regions (

D

i

⁢

(

A

)

are the Gerschgorin circles and

O

i

⁢

j

⁢

(

A

)

are the Brauer’s Cassini ovals); then

B

⁢

(

A

)

⊆

G

⁢

(

A

)

.

Proof:
Let

O

i

⁢

j

be one of the

n

⁢

(

n

-

1

)

/

2

ovals of Cassini for matrix

A

and be

z

∈

O

i

⁢

j

. If

R

i

=

0

or

R

j

=

0

, Brauer’s theorem imply

z

=

a

i

⁢

i

or

z

=

a

j

⁢

j

respectively; but since both

a

i

⁢

i

and

a

j

⁢

j

belong to their respective Gerschgorin circles, we have

z

∈

(

D

i

∪

D

j

)

. If both

R

i

>

0

and

R

j

>

0

, then we can write:

|

z

-

a

i

⁢

i

|

R

i

⋅

|

z

-

a

j

⁢

j

|

R

j

≤

1

.

For the left-hand side to be not greater than 1,

|

z

-

a

i

⁢

i

|

R

i

or

|

z

-

a

j

⁢

j

|

R

j

must be not greater than 1, which in turn means

z

∈

D

i

or

z

∈

D

j

, that is

z

∈

(

D

i

∪

D

j

)

. This way, we proved that

O

i

⁢

j

⊆

(

D

i

∪

D

j

)

; now, we have:

B

⁢

(

A

)

=

⋃

i

≠

j

O

i

⁢

j

⊆

⋃

i

=

1

n

D

i

=

G

⁢

(

A

)

.

3) It’s obvious from definition that there are infinitely many matrices which generate the same ovals of Cassini: namely, let’s define

Ω

⁢

(

A

)

=

{

M

∈

𝐂

n

×

n

:

m

i

⁢

i

=

a

i

⁢

i

,

R

i

⁢

(

M

)

=

R

i

⁢

(

A

)

}

as the set of all matrices which share the same ovals of Cassini as

A

. Then, by Brauer’s theorem, we have, for all

M

∈

Ω

matrices,

σ

⁢

(

M

)

⊆

B

⁢

(

A

)

,

and therefore, having defined

σ

⁢

(

Ω

)

=

⋃

M

∈

Ω

σ

⁢

(

M

)

, we have

σ

⁢

(

Ω

)

⊆

B

⁢

(

A

)

.

One may then ask how sharp this inclusion is, which, informally speaking, is

equivalent

to asking how ”efficient” is the ”use”, by Brauer’s theorem, of the 2n pieces of information

{

a

i

⁢

i

}

i

=

1

n

and

{

R

i

}

i

=

1

n

in the construction of inclusion sets (if for example we found the inclusion to be very loose, that is

σ

⁢

(

Ω

)

to be a very little subset of

B

⁢

(

A

)

, we could

conjecture

that the knowledge of the 2n numbers used by Brauer’s theorem should have led to a more precise bounding, since the spectra of all matrices which share these numbers lie in a much smaller region). It has been proven that actually

σ

⁢

(

Ω

)

=

B

⁢

(

A

)

,

thus showing Brauer’s ovals are

optimal ones

under this point of view.

References

1

S. Gerschgorin,

Uber die Abgrenzung der Eigenwerte einer Matrix

, Isv. Akad. Nauk USSR Ser. Mat., 7 (1931), pp. 749-754

2

A. Brauer,

Limits for the

characteristic roots

of a matrix II

, Duke Math. J. 14 (1947), pp. 21-26

3

R. S. Varga and A. Krautstengl,

On Gersgorin-type problems and ovals of Cassini

, Electron. Trans. Numer. Anal., 8 (1999), pp. 15-20

4

Richard S. Varga,

Gersgorin-type eigenvalue inclusion theorems and their sharpness

,Electronic Transactions on Numerical Analysis.
Volume 12 (2001), pp. 113-133

Title

Brauer’s ovals theorem

Canonical

name

BrauersOvalsTheorem

Date of creation

2013-03-22 15:35:30

Last modified on

2013-03-22 15:35:30

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

15

Author

Andrea Ambrosio (7332)

Entry type

Algorithm

Classification

msc 15A42

Related topic

GershgorinsCircleTheorem

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