characteristic matrix of diagonal element cross-section

characteristic matrix of diagonal element cross-section

Denote by

M

n

⁢

(

𝒦

)

the set of all

n

×

n

matrices over

𝒦

. Let

d

i

:

M

n

⁢

(

𝒦

)

⟶

𝒦

be
the function which extracts the

i

th

diagonal

element of a matrix.
Finally denote by

[

n

]

the set

{

1

,

…

,

n

}

.

Lemma

.

Let

K

be a field.
Let a sequence

A

1

,

…

,

A

r

∈

M

n

⁢

(

K

)

of

upper
triangular matrices

be given, and denote by

A

⊆

M

n

⁢

(

K

)

the unital

algebra

generated by these matrices.
For every sequence

λ

1

,

…

,

λ

r

∈

K

of
scalars there exists a matrix

C

∈

A

such that

d

i

⁢

(

C

)

=

{

1

if

d

i

⁢

(

A

k

)

=

λ

k

for all

k

∈

[

r

]

,

0

𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

for all

i

∈

[

n

]

.

A diagonal element of an upper triangular matrix is of course an

eigenvalue

of that matrix, so the particular

λ

1

,

…

,

λ

r

that one plugs into this lemma is typically either a sequence of
eigenvalues for the given matrices, or a sequence of values that
one thinks may be eigenvalues for these matrices.
The “cross-section” in the title refers to that there is one

λ

k

for each matrix

A

k

.

The result gets more interesting if one knows something more about

𝒜

than what was explicitly required above. A particular example
is that if the given matrices commute then

𝒜

will be a
commutative (

http://planetmath.org/Commutative

) algebra and consequently

C

will commute with all of the given matrices as well.

Proof.

Let

S

be the set of those row indices

i

for which

d

i

⁢

(

C

)

should be

0

, i.e.,

S

=

{

i

∈

[

n

]

|

d

i

⁢

(

A

k

)

≠

λ

k

for some

k

∈

[

r

]

}

⁢

,

and for each

i

∈

S

let

k

i

∈

[

r

]

be such that

d

i

⁢

(

A

k

i

)

≠

λ

k

i

. Define

B

i

=

{

A

k

i

-

d

i

⁢

(

A

k

i

)

⁢

I

if

i

∈

S

,

I

otherwise

for all

i

∈

[

n

]

and let

B

=

∏

i

=

1

n

B

i

. Since all
the matrices involved are upper triangular, a diagonal element in

B

is simply the product of the corresponding diagonal elements in all
of

B

1

,

…

,

B

n

. If

i

∈

S

then

d

i

⁢

(

B

i

)

=

d

i

⁢

(

A

k

i

-

d

i

⁢

(

A

k

i

)

⁢

I

)

=

d

i

⁢

(

A

k

i

)

-

d

i

⁢

(

A

k

i

)

=

0

and thus

d

i

⁢

(

B

)

=

0

for

i

∈

S

. If instead

i

∈

[

n

]

∖

S

then

d

i

⁢

(

B

)

=

∏

j

=

1

n

d

i

⁢

(

B

i

)

=

∏

j

∈

S

d

i

⁢

(

A

k

j

-

d

j

⁢

(

A

k

j

)

⁢

I

)

=

=

∏

j

∈

S

(

d

i

(

A

k

j

)

-

d

j

(

A

k

j

)

)

=

∏

j

∈

S

(

λ

k

j

-

d

j

(

A

k

j

)

)

=

:

b

.

This

b

is by

definition

of

k

j

nonzero, and since it is
independent of

i

it follows that it is the only nonzero value of a
diagonal element of

B

. Hence the wanted

C

=

b

-

1

⁢

B

.
∎

Title

characteristic matrix of diagonal element cross-section

Canonical

name

CharacteristicMatrixOfDiagonalElementCrosssection

Date of creation

2013-03-22 15:29:45

Last modified on

2013-03-22 15:29:45

Owner

lars_h (9802)

Last modified by

lars_h (9802)

Numerical id

4

Author

lars_h (9802)

Entry type

Theorem

Classification

msc 15A30

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