Page 23 - Matrices theory and applications

1. Elementary Theory
6
1.1.3 Product of Matrices
Let n, m, p ≥ 1 be three positive integers. We define a (noncommutative)
multiplication law
M n×m (K) × M m×p (K) → M n×p (K),

MM ,
(M, M ) →

which we call the product of M and M .The matrix M = MM is given

by the formula
m

m = m ik m , 1 ≤ i ≤ n, 1 ≤ j ≤ p.

ij kj
k=1
We check easily that this law is associative: if M, M ,and M have

respective sizes n × m, m × p, p × q,one has
(MM )M = M(M M ).

The product is distributive with respect to addition:

M(M + M )= MM + MM , (M + M )M = MM + M M .

It also satisfies

a(MM )= (aM)M = M(aM ), ∀a ∈ K.

Last, if m = n,then I n M = M . Similarly, if m = p,then MI m = M.
The product is an internal composition law in M n (K), which endows
this space with a structure of a unitary K-algebra. It is noncommutative
in general. For this reason, we define the commutator of M and N by
[M, N]:= MN − NM. For a square matrix M ∈ M n (K), one defines
2
2
3
k
2
M = MM, M = MM = M M (from associativity), ..., M k+1 = M M.
0
j
k
1
One completes this notation by M = M and M = I n . One has M M =
k
M j+k for all j, k ∈ IN.If M = 0 for some integer k ∈ IN,one says that
M is nilpotent.One says that M is idempotent if I n − M is nilpotent.
One says that two matrices M, N ∈ M n (K) commute with each other
if MN = NM. The powers of a square matrix M commute pairwise. In
particular , the set K(M) formed by polynomials in M, which cinsists of
matrices of the form
r
a 0 I n + a 1 M + ··· + a r M , a 0 ,... ,a r ∈ K, r ∈ IN,
is a commutative algebra.
One also has the formula (see Exercise 2)
rk(MM ) ≤ min{rk M, rk M }.

1.1.4 Matrices as Linear Maps
Let E, F be two K-vector spaces. A map u : E → F is linear (one also
speaks of a homomorphism)if u(x + y)= u(x)+ u(y)and u(ax)= au(x)