Page 26 - Matrices theory and applications

1.2. Change of Basis
9

If E = F and u ∈ End(E), one may compare the matrices M, M of u

in two different bases β, β (here γ = β and γ = β ). The above formula
becomes
−1
M = P
MP.

One says that M and M are similar, or that they are conjugate (the latter
term comes from group theory). One also says that M is the conjugate of

M by P.
The equivalence and the similarity of matrices are two equivalence
relations. They will be studied in Chapter 6.
1.2.1 Block Decomposition
Considering matrices with entries in a ring A does not cause difficulties, as
long as one limits oneself to addition and multiplication. However, when A
is not commutative, it is important to choose the formula
m

M ij M jk
j=1
when computing (MM ) ik , since this one corresponds to the composition

n
law when one identifies matrices with A-linear maps from A m to A .
When m = n, the product is a composition law in M n (K). This space
is thus a K-algebra. In particular, it is a ring, and one may consider the
matrices with entries in B = M n (K). Let M ∈ M p×q (B)have entries M ij
(one chooses uppercase letters in order to keep in mind that the entries

are themselves matrices). One naturally identifies M with the matrix M ∈
M pn×qn (K), whose entry of indices ((i − 1)n + k, (j − 1)n + l), for i ≤ p,
j ≤ q,and k, l ≤ n, is nothing but
(M ij ) kl .
One verifies easily that this identification is an isomorphism between
M p×q (B)and M pn×qn (K)as K-vector spaces.
More generally, choosing decompositions n = n 1 +···+n r , m = m 1 +···+
m s with n k ,m l ≥ 1, one may associate to every matrix M ∈ M n×m (K)
˜
an array M with r rows and s columns whose element of index (k, l)is a
˜
matrix M kl ∈ M n k ×m l (K). Defining

ν k = n t , µ l = m t (ν 1 = µ 1 =0),
t<k t<l
one has by definition
˜
(M kl ) ij = m ν k +i,µ l +j , 1 ≤ i ≤ n k , 1 ≤ j ≤ m l .
This procedure, which depends on the choice of n k ,m l , is called block
decomposition.