Page 48 - Matrices theory and applications

2.9. Exercises
31
otherwise. Saying that a matrix is reducible is equivalent to saying that
−1
is of block-triangular
there exists a permutation matrix P such that PAP
form

B
C
,
D
0 p,n−p
with 1 ≤ p ≤ n − 1. As a matter of fact, P is the matrix of the transforma-
tion from a basis γ to the canonical one, γ being obtained by first writing
j
the vectors e with j ∈ J, and then those with j ∈ I.Working in thenew
basis amounts to decomposing the linear system Ax = b into two subsys-
tems Dz = d and By = c − Cz, which are to be solved successively. The
spectrum of A is the union of those of B and D, so that many interesting
questions concerning square matrices reduce to questions about irreducible
matrices.
We shall see in the exercises a characterization of irreducible matrices in
terms of graphs. Here is a useful consequence of irreducibility.
Proposition 2.8.1 Let M ∈ M n (K) be an irreducible matrix such that
i ≥ j +2 implies m ij =0. Then the eigenvalues of M are geometrically
simple.
Proof
The hypothesis implies that all entries m i+1,i are nonzero. If λ is an eigen-
¯
value, let us consider the matrix N ∈ M n−1 (K), obtained from M − λI n
by deleting the first row and the last column. It is a triangular matrix,
whose diagonal terms are nonzero. It is thus invertible, which implies
rk(M − λI n )= n − 1. Hence ker(M − λI n ) is of dimension one.
2.9 Exercises

1. Verify that the product of two triangular matrices of the same type
(upper or lower) is triangular, of the same type.

2. Prove in full detail that the determinant of a triangular matrix (re-
spectively a block-triangular one) equals the product of its diagonal
terms (respectively the product of the determinants of its diagonal
blocks).

3. Find matrices M, N ∈ M 2 (K)suchthat MN =0 2 and NM =0 2 .
Such an example shows that MN and NM are not necessarily similar,
though they would be in the case where M or N is invertible.
4. Characterize the square matrices that are simultaneously orthogonal
and triangular.