simultaneous upper triangular block-diagonalization of commuting matrices

simultaneous upper triangular block-diagonalization of commuting matrices

Let

𝐞

i

denote the (column) vector whose

i

th position is

1

and where all other positions are

0

. Denote by

[

n

]

the set

{

1

,

…

,

n

}

. Denote by

M

n

⁢

(

𝒦

)

the set of all

n

×

n

matrices over

𝒦

, and by

GL

n

⁢

(

𝒦

)

the set of all

invertible elements

of

M

n

⁢

(

𝒦

)

.

Theorem 1

.

Let

K

be a field, let

A

1

,

…

,

A

r

∈

M

n

⁢

(

K

)

be pairwise

commuting matrices

, and let

L

be a

field extension

of

K

in which the

characteristic polynomials

of all

A

k

split. Then there exists an

equivalence relation

∼

on

[

n

]

and
a matrix

R

∈

GL

n

⁢

(

L

)

such that:

1.

If

i

∼

j

and

i

⩽

k

⩽

j

then

k

∼

i

.

2.

If

i

∼

j

then

𝐞

i

T

⁢

R

-

1

⁢

A

k

⁢

R

⁢

𝐞

i

=

𝐞

j

T

⁢

R

-

1

⁢

A

k

⁢

R

⁢

𝐞

j

.

3.

If

𝐞

i

T

⁢

R

-

1

⁢

A

k

⁢

R

⁢

𝐞

j

≠

0

then

i

⩽

j

and

i

∼

j

.

In other words there exists a simultaneous

upper triangular

block-diagonalisation of the matrices

A

1

,

…

,

A

r

in which each
block is characterised by the particular values of the

diagonal

elements

.

The proof of this

theorem

is the obvious

combination

of the following
two lemmas.

Lemma 2

.

Let

K

be a field, let

A

1

,

…

,

A

r

∈

M

n

⁢

(

K

)

be pairwise commuting matrices, and let

L

be a field extension
of

K

in which the characteristic polynomials of all

A

k

split. Then there exists some

P

∈

GL

n

⁢

(

L

)

such that

1.

P

-

1

⁢

A

k

⁢

P

is upper triangular for all

k

=

1

,

…

,

r

,
and

2.

if

i

,

j

,

l

∈

[

n

]

are such that

i

⩽

l

⩽

j

and

𝐞

i

T

⁢

P

-

1

⁢

A

k

⁢

P

⁢

𝐞

i

=

𝐞

j

T

⁢

P

-

1

⁢

A

k

⁢

P

⁢

𝐞

j

for all

k

=

1

,

…

,

r

, then

𝐞

l

T

⁢

P

-

1

⁢

A

k

⁢

P

⁢

𝐞

l

=

𝐞

j

T

⁢

P

-

1

⁢

A

k

⁢

P

⁢

𝐞

j

for all

k

=

1

,

…

,

r

as well.

Let

B

k

=

P

-

1

⁢

A

k

⁢

P

for all

k

=

1

,

…

,

r

and define

i

∼

j

if and only if

𝐞

i

T

⁢

P

-

1

⁢

A

k

⁢

P

⁢

𝐞

i

=

𝐞

j

T

⁢

P

-

1

⁢

A

k

⁢

P

⁢

𝐞

j

⁢

for all

⁢

k

∈

[

r

]

⁢

.

Lemma 3

.

Let

L

be a field, let

n

be a

positive integer

, and let

∼

be an equivalence relation on

[

n

]

such that if

i

∼

j

and

i

⩽

k

⩽

j

then

k

∼

i

. Let

B

1

,

…

,

B

r

∈

M

n

⁢

(

L

)

be pairwise commuting

upper triangular matrices

.
If these matrices and

∼

are related such that

i

∼

j

if and only if

𝐞

i

T

⁢

B

k

⁢

𝐞

i

=

𝐞

j

T

⁢

B

k

⁢

𝐞

j

⁢

for all

⁢

k

∈

[

r

]

⁢

,

then there exists a matrix

Q

∈

GL

n

⁢

(

L

)

such that:

1.

If

𝐞

i

T

⁢

Q

-

1

⁢

B

k

⁢

Q

⁢

𝐞

j

≠

0

then

i

∼

j

and

i

⩽

j

.

2.

If

i

∼

j

then

𝐞

i

T

⁢

Q

-

1

⁢

B

k

⁢

Q

⁢

𝐞

j

=

𝐞

i

T

⁢

B

k

⁢

𝐞

j

.

The wanted

R

is then

P

⁢

Q

.

Title

simultaneous upper triangular block-diagonalization of commuting matrices

Canonical name

SimultaneousUpperTriangularBlockdiagonalizationOfCommutingMatrices

Date of creation

2013-03-22 15:29:35

Last modified on

2013-03-22 15:29:35

Owner

lars_h (9802)

Last modified by

lars_h (9802)

Numerical id

4

Author

lars_h (9802)

Entry type

Theorem

Classification

msc 15A21

Related topic

JordanCanonicalFormTheorem

Related topic

IfABInM_nmathbbCABBAThenQHAQT_1QHBQT_2

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