simultaneous upper triangular block-diagonalization of commuting matrices
Let πi denote the (column) vector whose ith position is 1 and where all other positions are 0. Denote by [n] the set {1,β¦,n}. Denote by Mn(π¦) the set of all nΓn matrices over π¦, and by GLn(π¦) the set of all invertible elements of Mn(π¦).
Theorem 1.
Let K be a field, let A1,β¦,ArβMn(K)
be pairwise commuting matrices, and let L be a field extension
of K in which the characteristic polynomials
of all Ak
split. Then there exists an equivalence relation
βΌ on [n] and
a matrix RβGLn(L) such that:
-
1.
If iβΌj and iβ©½ then .
-
2.
If then .
-
3.
If then and .
In other words there exists a simultaneous upper triangular block-diagonalisation of the matrices 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 be a field, let be pairwise commuting matrices, and let be a field extension of in which the characteristic polynomials of all split. Then there exists some such that
-
1.
is upper triangular for all , and
-
2.
if are such that and for all , then for all as well.
Let for all and define
Lemma 3.
Let be a field, let be a positive integer, and let be an equivalence relation on such that if and then . Let be pairwise commuting upper triangular matrices. If these matrices and are related such that
then there exists a matrix such that:
-
1.
If then and .
-
2.
If then .
The wanted is then .
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 |