Page 22 - Matrices theory and applications

5
1.1. Basics
consecutive numbers. One needs only two finite sets, one for indexing the
rows, the other for indexing the columns.
The set of matrices of size n × m with entries in K is denoted by

M n×m(K). It is an additive group, where M + M denotes the matrix M

whose entries are given by m = m ij + m . One defines likewise multipli-

ij
ij

cation by a scalar a ∈ K.The matrix M := aM is defined by m = am ij .

ij
One has the formulas a(bM)= (ab)M, a(M + M )= (aM)+(aM ), and

(a + b)M =(aM)+(bM), which endow M n×m (K)witha K-vector space
structure. The zero matrix is denoted by 0, or 0 nm when one needs to avoid
ambiguity.
When m = n, one writes simply M n (K) instead of M n×n(K), and 0 n
instead of 0 nn . The matrices of sizes n × n are called square matrices. One
writes I n for the identity matrix, defined by

j 0, if i = j,
m ij = δ =
i 1, if i = j.
In other words,
 
1 0 ··· 0
. .
 . . . 
 0 . . . . 
I n =  .  .

 . . . . . . . . 0 

0 ··· 0 1
The identity matrix is a special case of a permutation matrix,which are
square matrices having exactly one nonzero entry in each row and each
column, that entry being a 1. In other words, a permutation matrix M
reads
σ(j)
m ij = δ
i
for some permutation σ ∈ S n .
A square matrix for which i< j implies m ij = 0 is called a lower
triangular matrix. It is upper triangular if i> j implies m ij =0. It is
strictly upper triangular if i ≥ j implies m ij = 0. Last, it is diagonal if m ij
vanishes for every pair (i, j)such that i = j. In particular, given n scalars
d 1 ,... ,d n ∈ K, one denotes by diag(d 1 ,... ,d n ) the diagonal matrix whose
diagonal term m ii equals d i for every index i.
When m =1, a matrix M of size n × 1 is called a column vector.One
identifies it with the vector of K n whose ith coordinate in the canonical
basis is m i1 . This identification is an isomorphism between M n×1 (K)and
n
K . Likewise, the matrices of size 1 × m are called row vectors.
Amatrix M ∈ M n×m(K) may be viewed as the ordered list of its
columns M (j) (1 ≤ j ≤ m). The dimension of the linear subspace spanned
n
by the M (j) in K is called the rank of M and denoted by rk M.