14.17 The Spectral Norm And The Operator Norm
Another way of describing a norm for an n × n matrix is as follows.
Definition 14.17.1 Let A be an m × n matrix. Define the spectral norm of A, written as
That is, the largest singular value of A. (Note the eigenvalues of A∗A are all positive because if A∗Ax = λx,
Actually, this is nothing new. It turns out that
is nothing more than the operator norm for A
taken with respect to the usual Euclidean norm,
Proposition 14.17.2 The following holds.
Proof: Note that A∗A is Hermitian and so by Corollary 14.11.6,
Here is another proof of this proposition. Recall there are unitary matrices of the right size U,V such
that A = U
where the matrix on the inside is as described in the section on the singular
value decomposition. Then since unitary matrices preserve norms,
This completes the alternate proof.
From now on,
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.