simultaneous upper triangular block-diagonalization of commuting matrices
Let denote the (column) vector whose th position is and where all other positions are . Denote by the set . Denote by the set of all matrices over , and by the set of all invertible elements of .
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 an equivalence relation on and a matrix such that:
If and then .
If then .
If then and .
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
is upper triangular for all , and
if are such that and for all , then for all as well.
Let for all and define
The wanted is then .
|Title||simultaneous upper triangular block-diagonalization of commuting matrices|
|Date of creation||2013-03-22 15:29:35|
|Last modified on||2013-03-22 15:29:35|
|Last modified by||lars_h (9802)|