13.19. THE SPECTRAL NORM AND THE OPERATOR NORM 381
which says
A0 =V
(σ−1 0
0 0
)U∗ ≡ A+. ■
The theorem is significant because there is no mention of eigenvalues or eigenvectors inthe characterization of the Moore Penrose inverse given in 13.36. It also shows immediatelythat the Moore Penrose inverse is a generalization of the usual inverse. See Problem 3.
13.19 The Spectral Norm And The Operator NormAnother way of describing a norm for an n×n matrix is as follows.
Definition 13.19.1 Let A be an m× n matrix. Define the spectral norm of A, written as||A||2 to be
max{
λ1/2 : λ is an eigenvalue of A∗A
}.
That is, the largest singular value of A. (Note the eigenvalues of A∗A are all positivebecause if A∗Ax= λx, then
λ |x|2 = λ (x,x) = (A∗Ax,x) = (Ax,Ax)≥ 0.)
Actually, this is nothing new. It turns out that ||·||2 is nothing more than the operatornorm for A taken with respect to the usual Euclidean norm,
|x|=
(n
∑k=1|xk|2
)1/2
.
Proposition 13.19.2 The following holds.
||A||2 = sup{|Ax| : |x|= 1} ≡ ||A|| .
Proof: Note that A∗A is Hermitian and so by Corollary 13.13.6,
||A||2 = max{(A∗Ax,x)1/2 : |x|= 1
}= max
{(Ax,Ax)1/2 : |x|= 1
}= max{|Ax| : |x|= 1}= ||A|| . ■
Here is another proof of this proposition. Recall there are unitary matrices of the right
size U,V such that A = U
(σ 00 0
)V ∗ where the matrix on the inside is as described
in the section on the singular value decomposition. Then since unitary matrices preservenorms,
||A|| = sup|x|≤1
∣∣∣∣∣U(
σ 00 0
)V ∗x
∣∣∣∣∣= sup|V ∗x|≤1
∣∣∣∣∣U(
σ 00 0
)V ∗x
∣∣∣∣∣= sup
|y|≤1
∣∣∣∣∣U(
σ 00 0
)y
∣∣∣∣∣= sup|y|≤1
∣∣∣∣∣(
σ 00 0
)y
∣∣∣∣∣= σ1 ≡ ||A||2
This completes the alternate proof.From now on, ||A||2 will mean either the operator norm of A taken with respect to the
usual Euclidean norm or the largest singular value of A, whichever is most convenient.