Page 49 - Matrices theory and applications

2. Square Matrices
32
2
= M a projection
5. One calls any square matrix M satisfying M
matrix,or projector.
(a) Let P ∈ M n(K) be a projector, and let E =ker P, F =ker(I n −
n
P). Show that K = E ⊕ F.
(b) Let P, Q be two projectors. Show that (P − Q) commute with
P and with Q. Also, prove the identity
2
2
(P − Q) +(I n − P − Q) = I n . 2
6. Let M be a square matrix over a field K, which we write blockwise
as

A B
M = .
C D
The formula det M =det(AD−BC) is meaningless in general, except
when A, B, C, D have the same size. In that case the formula is false,
with the exception of scalar blocks. Compare with Schur’s formula
(Proposition 8.1.2).
7. If A, B, C, D ∈ M m (K)and if AC = CA, show that the determinant
of

A B
M =
C D
equals det(AD − CB). Begin with the case where A is invertible, by
computing the product

I m 0 m
M.
−C A
¯
Then apply this intermediate result to the matrix A−zI n ,with z ∈ K
a suitable scalar.
Compare with the previous exercise.
8. Verify that the inverse of a triangular matrix, whenever it exists, is
triangular of the same type.
9. Show that the eigenvalues of a triangular matrix are its diagonal
entries. What are their algebraic multiplicities?

10. Let A ∈ M n (K) be given. One says that a list (a 1σ(1) ,... ,a nσ(n) )
is a diagonal of A if σ is a permutation (in that case, the diagonal
given by the identity is the main diagonal). Show the equivalence of
the following properties.

• Every diagonal of A contains a zero element.
• There exists a null matrix extracted from A of size k × l with
k + l> n.