Page 82 - Matrices theory and applications

4.1. A Brief Review
65
Equivalently,
A =sup Ax =max Ax .
x ≤1
x ≤1
One verifies easily that A → A is a norm on M n (K). It is called the norm
induced by that of E,or the norm subordinated to that of E.Thoughwe
adopted the same notation · for the two norms, that on E and that on
M n (K), these are, of course, distinct objects. In many places, one finds the
notation ||| · ||| for the induced norm. When one does not wish to mention
from which norm on E agiven norm on M n (K) is induced, one says that
A → A is a matrix norm. The main properties of matrix norms are
AB ≤ A B , I n =1.
These properties are those of any algebra norm (otherwise called norm of
k
k
algebra, see Section 4.4). In particular, one has A ≤ A for every
k ∈ IN.
p
Here are a few examples induced by the norms l :
i=n

= max |a ij |,
A 1
1≤j≤n
i=1
j=n

= max |a ij |,
A ∞
1≤i≤n
j=1
∗
A 2 = ρ(A A) 1/2 .
To prove these formulas, we begin by proving the inequalities ≥, selecting
a suitable vector x, and writing A p ≥ Ax p / x p.For p =1 we choose
an index j such that the maximum in the above formula is achieved. Then
we let x j = 1, while x k =0 otherwise. For p = ∞,welet x j =¯a i 0 j /|a i 0 j |,
where i 0 achieves the maximum in the above formula; For p =2 we choose
an eigenvector of A A associated to an eigenvalue of maximal modulus.
∗
We thus obtain three inequalities. The reverse inequalities are direct con-
sequences of the definitions. The values of A 1 and A ∞ illustrate a
particular case of the general formula
(y Ax)
∗
A = A =sup sup .
∗

x =0 y =0 x · y
Proposition 4.1.5 For an induced norm, the condition B < 1 implies
that I n − B is invertible, with inverse given by the sum of the series
∞
k
B .
k=0
Proof
k
k
k
The series B is normally convergent, since B ≤ B ,
k k k
where the latter series converges because B < 1. Since M n(K)is com-