ne
previo
Next:
Direct Methods of Solving
Up:
Norms and Convergence
Previous:
Norms of Vectors
Because we deal with equations like
which involve a matrix as well as vectors, we extend norms
to matrices. We could of course regard
as a vector (with some order specified) in
, but since
matrix multiplication may arise, we add an extra property.
Definition 2.2.1
A
matrix norm
on
is a mapping
with the properties:
(i)
,
(ii)
,
(iii)
,
(iv)
.
Not all vector norms will become matrix norms in
because of axiom (iv). Most of the norms we use here
are derived from vector norms in the following way:
Definition 2.2.2
The matrix norm
induced by
or
subordinate to
the given vector norm
is given by
(2.1)
We can regard
as the maximum `magnification' capability of
A
.
Note that replacing
by
in this definition makes no difference; if we choose
so that
, we obtain the alternative definition
Matrix Norms
http://www.maths.lancs.ac.uk/~gilbert/m306a/node6.html
1 of 10
10/22/10 11:43 PM
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This sup is attained if
is a continuous function of the components of
, since the surface of the unit
ball
is compact.
Define
by
; then we show that
is a vector seminorm.
(i)
. Unfortunately
does not imply that
, so
is not a
norm, but a seminorm. However a seminorm possesses the bound attainment property, so we can
work with this.
(ii)
.
(iii)
.
Thus
is a seminorm and hence
attains its maximum on the unit ball. We may now write the
last definition in the form
(2.2)
Finally, when a lot of manipulation is to be performed, we can write the last definition as
where
is the value of
in the previous definition at which the maximum is attained. Clearly
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Chandrasekara
 Linear Algebra, Matrices, Hermitian, matrix norms

Click to edit the document details