Block Diagonalization
126 (2001)
MATHEMATICA BOHEMICA
No. 1, 237–246
BLOCK DIAGONALIZATION
J. J. Koliha, Melbourne
(Received June 15, 1999)
Abstract. We study block diagonalization of matrices induced by resolutions of the unit matrix into the sum of idempotent matrices. We show that the block diagonal matrices have disjoint spectra if and only if each idempotent matrix in the inducing resolution double commutes with the given matrix. Applications include a new characterization of an eigenprojection and of the Drazin inverse of a given matrix.
Keywords : eigenprojection, resolutions of the unit matrix, block diagonalization
MSC 2000 : 15A21, 15A27, 15A18, 15A09
1. Introduction and preliminaries
In this paper we are concerned with a block diagonalization of a given matrix A; by definition, A is block diagonalizable if it is similar to a matrix of the form
(1.1)
A1
0
...
0
0
A2
0
...
0
...
0
= diag(A1 , . . . , Am ).
...
. . . Am
Then the spectrum σ (A) of A is the union of the spectra σ (A1 ), . . . , σ (Am ), which in general need not be disjoint. The ultimate such diagonalization is the Jordan form of the matrix; however, it is often advantageous to utilize a coarser diagonalization, easier to construct, and customized to a particular distribution of the eigenvalues.
The most useful diagonalizations are the ones for which the sets σ (Ai ) are pairwise disjoint; it is the aim of this paper to give a full characterization of these diagonalizations.
For any matrix A ∈ n×n we denote its kernel and image by ker A and im A, respectively. A matrix E is idempotent (or a projection matrix ) if E 2 = E , and
237
nilpotent if E p = 0 for some positive integer p. Recall that rank E = tr E if E is idempotent. Matrices A, B are similar, written A ∼ B , if A = Q−1 BQ for some nonsingular matrix Q. The similarity transformation will be also written explixitly as ψ (U ) = ψQ (U ) = Q−1 U Q. The commutant and the double
MATHEMATICA BOHEMICA
No. 1, 237–246
BLOCK DIAGONALIZATION
J. J. Koliha, Melbourne
(Received June 15, 1999)
Abstract. We study block diagonalization of matrices induced by resolutions of the unit matrix into the sum of idempotent matrices. We show that the block diagonal matrices have disjoint spectra if and only if each idempotent matrix in the inducing resolution double commutes with the given matrix. Applications include a new characterization of an eigenprojection and of the Drazin inverse of a given matrix.
Keywords : eigenprojection, resolutions of the unit matrix, block diagonalization
MSC 2000 : 15A21, 15A27, 15A18, 15A09
1. Introduction and preliminaries
In this paper we are concerned with a block diagonalization of a given matrix A; by definition, A is block diagonalizable if it is similar to a matrix of the form
(1.1)
A1
0
...
0
0
A2
0
...
0
...
0
= diag(A1 , . . . , Am ).
...
. . . Am
Then the spectrum σ (A) of A is the union of the spectra σ (A1 ), . . . , σ (Am ), which in general need not be disjoint. The ultimate such diagonalization is the Jordan form of the matrix; however, it is often advantageous to utilize a coarser diagonalization, easier to construct, and customized to a particular distribution of the eigenvalues.
The most useful diagonalizations are the ones for which the sets σ (Ai ) are pairwise disjoint; it is the aim of this paper to give a full characterization of these diagonalizations.
For any matrix A ∈ n×n we denote its kernel and image by ker A and im A, respectively. A matrix E is idempotent (or a projection matrix ) if E 2 = E , and
237
nilpotent if E p = 0 for some positive integer p. Recall that rank E = tr E if E is idempotent. Matrices A, B are similar, written A ∼ B , if A = Q−1 BQ for some nonsingular matrix Q. The similarity transformation will be also written explixitly as ψ (U ) = ψQ (U ) = Q−1 U Q. The commutant and the double