simultaneous block-diagonalization of upper triangular 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 the function which extracts the th diagonal element of a matrix, i.e., .
Condition 1 says that if an element of is nonzero then both its row and column indices must belong to the same equivalence class of , i.e., the nonzero elements of only occur in particular blocks (http://planetmath.org/PartitionedMatrix) along the diagonal, and these blocks correspond to equivalence classes of . Condition 2 says that within one of these blocks, is equal to .
The proof of the theorem requires the following lemma.
The proof of that lemma can be found in this article (http://planetmath.org/CharacteristicMatrixOfDiagonalElementCrossSection).
Proof of theorem.
The proof is by induction on the number of equivalence classes of . If there is only one equivalence class then one can take .
If there is more than one equivalence class, then let be the equivalence class that contains . By Lemma 2 there exists a matrix in the unital algebra generated by (hence necessarily upper triangular) such that for all and for all . Thus has a
where is a matrix that has all s on the diagonal, and is a matrix that has all s on the diagonal.
Let be arbitrary and similarly decompose
One can identify and , but due to the zero diagonal of and the fact that the of these matrices are smaller than , the more striking equality also holds. As for , one may conclude that it is invertible.
Since the algebra that belongs to was generated by pairwise commuting elements, it is a commutative (http://planetmath.org/Commutative) algebra, and in particular . In of the individual blocks, this becomes
and consider the matrix . Clearly
so that the positions with row not in and column in are all zero, as requested for . It should be observed that the choice of is independent of , and that the same thus works for all the .
In order to complete the proof, one applies the induction hypothesis to the restriction of to and the corresponding submatrices of , which satisfy the same conditions but have one equivalence class less. This produces a block-diagonalising matrix for these submatrices, and thus the sought can be constructed as . ∎
|Title||simultaneous block-diagonalization of upper triangular commuting matrices|
|Date of creation||2013-03-22 15:29:42|
|Last modified on||2013-03-22 15:29:42|
|Last modified by||lars_h (9802)|