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 commutant of a matrix A ∈ n×n are deﬁned by

comm(A) = {U ∈

2

comm (A) = {U ∈

n×n

: AU = U A},

n×n

: U V = V U for all V ∈ comm(A)}.

The main result of the present paper is the fact that the block diagonal matrices have disjoint spectra if and only if the idempotent matrices inducing the diagonalization double commute with the given matrix. One way to prove this is to use a powerful theorem of matrix theory which states that the second commutant comm2 (A) coincides with the set of all matrices of the form f (A), where f is a polynomial ([10, Theorem 1.3.7] or [11, p. 106]). (A simple proof of this result is given by Lagerstrom in [4].) We prefer to give an elementary proof of the disjoint spectra theorem, which provides us with a greater insight. As a corollary we show that an idempotent matrix E is an eigenprojection of A if and only if it is in the double commutant of A and (A − µI )E is nilpotent for some µ.

To study block diagonalizations of the form (1.1), it is convenient to use resolutions of the unit matrix whose components are idempotent matrices. Deﬁniton 1.1. Let m 2 be an integer. An m-tuple (E1 , . . . , Em ) of idempotent n × n matrices is called a resolution of the unit matrix if E1 + . . . + Em = I . A resolution is called nontrivial if each component matrix is nonzero. We say that the resolution (E1 , . . . , Em ) commutes with the matrix A if Ei ∈ comm(A) for each i, and that it double commutes with A if Ei ∈ comm2 (A) for each i.

ÆÓØ

1.2. If (E1 , . . . , Em ) is a nontrivial resolution of the unit n × n matrix, then n = tr I = tr(E1 + . . . + Em ) = tr E1 + . . . + tr Em = rank E1 + . . . + rank Em , which means that im Ei ∩ im Ej = {0} if i = j . Therefore

(1.2)

Ei Ej = 0 if i = j.

We will need the following result on projections associated with direct sums of subspaces, which can be deduced from [7, Theorem 214B].

Lemma 1.3. n is the direct sum n = X1 ⊕ . . . ⊕ Xm of nonzero subspaces Xi if and only if there is a nontrivial...