HP MLIB User's Guide Vol. 2 7th Ed.

Chapter 9 LAPACK Auxiliary Subprograms 653
What you need to know to use these subprograms
Definition: A matrix norm on R
m×n
, the vector space of m-by-n real matrices, is
a function f:R
m×n
→ R that has the following properties:
As with vector norms, f is denoted with the double-bar notation: f(A)=||A||,
again using subscripts to designate different matrix norms.
The formal definition of a matrix norm, given above, ignores the uses of
matrices as operators in matrix-vector and matrix-matrix multiplication.
Therefore, a matrix norm usually is required to satisfy several additional
conditions related to such products.
Let || ||
α
be a vector norm on R
m
, || ||
β
be a vector norm on R
n
, and || ||
α,β
be a
matrix norm on R
m×n
. The matrix norm is said to be consistent with the vector
norm if
Let || ||
α
be a vector norm on R
m
, || ||
β
be a vector norm on R
n
, and || ||
α,β
be a
matrix norm on R
m×n
. The matrix norm is said to be induced by or subordinate
to the vector norm if
Finally, let || ||
α
, || ||
β
, and|| ||
γ
be matrix norms on R
m×q
, R
m×n
, and R
n×q
,
respectively. The norms are consistent if the submultiplicative property
is satisfied for all A ε R
m×n
and B ε R
n×q
.
f(A) ≥ 0
A ε R
m×n
f(A) = 0 if and only if A =0
f(A+B) ≤ f(A)+f(B)
A, B ε R
m×n
f(αA) = |α|f(A)
αεR, A ε R
m×n
Ax
β
A
a β,
x
a.
≤
A
a β,
max
x 0≠
=
Ax
β
x
a
---------------
.
AB
α
A
β
B
γ
≤