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Direct Methods of Solving
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Norms and Convergence
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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
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10/22/10 11:43 PM
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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
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This note was uploaded on 01/14/2011 for the course ECE 210a taught by Professor Chandrasekara during the Fall '08 term at UCSB.
 Fall '08
 Chandrasekara

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