Page 27 - Matrices theory and applications

1. Elementary Theory
10
˜
Though M is not strictly speaking a matrix (except in the case studied
previously where the n k ,m l are all equal to each other), one still may define
˜
the sum and the product of such objects. Concerning the product of M and
˜
M , we must of course be able to compute the products M jk M , and thus
˜
˜
kl
the sizes of blocks must be compatible. One verifies easily that the block
decomposition behaves well with respect to the addition and the product.
For instance, if n = n 1 + n 2 , m = m 1 + m 2 and p = p 1 + p 2 , two matrices

M, M of sizes n × m and m × p, with block decomposition M ij , M ,have

kl

a product M = MM ∈ M n×p (K), whose block decomposition M ij is

given by

M = M i1 M + M i2 M .
ij 1j 2j
A square matrix M, whose block decomposition is the same according to
rows and columns (that is m k = n k , in particular the diagonal blocks are
square matrices) is said lower block-triangular if the blocks M kl with k< l
are null blocks. One defines similarly the upper block-triangular matrices or
the block-diagonal matrices.
1.2.2 Transposition
If M ∈ M n×m(K), one defines the transposed matrix of M (or simply the
transpose of M)by
M T =(m ji ) 1≤i≤m,1≤j≤n .
The transposed matrix has size m × n,andits entries ˆm ij are given by
T

ˆ m ij = m ji . When the product MM makes sense, one has (MM ) =
T
(M ) M T (note that the orders in the two products are reversed). For two
T
T
T
matrices of the same size, (M + M ) = M +(M ) . Finally, if a ∈ K,
T
T

then (aM) = a(M ). The map M → M defined on M n (K) isthuslinear,
but it is not an algebra endomorphism.
A matrix and its transpose have the same rank. A proof of this fact is
given at the end of this section.
T
T
For every matrix M ∈ M n×m (K), the products M M and MM always
make sense. These products are square matrices of sizes m × m and n × n,
respectively.
A square matrix is said to be symmetric if M T = M,and skew-symmetric
if M T = −M (notice that these two notions coincide when K has char-
T
acteristic 2). When M ∈ M n×m (K), the matrices M M and MM T are
symmetric. Wedenoteby Sym (K) the subset of symmetric matrices in
n
M n (K). It is a linear subspace of M n (K). The product of two symmetric
matrices need not be symmetric.
T
A square matrix is called orthogonal if M M = I n .We shall seein
Section 2.2 that this condition is equivalent to MM T = I n .
T
n
m
If M ∈ M n×m (K), y ∈ K ,and x ∈ K , then the product x My
T
T
belongs to M 1 (K) and is therefore a scalar, equal to y M x.Saying that