Norms of Matrices
We can measure matrix sizes using vector norms, because
R
m
×
n
is a vector space.
Although the names are different, all of the pnorms above give matrix norms if the
matrix is stretched into one long vector. The only matrix norm of this flavor that is
used often is the Frobenius norm
k
A
k
F
= (
m
X
i
=1
n
X
j
=1

a
ij

2
)
1
/
2
= tr(
A
t
A
)
1
/
2
.
It is the vector 2norm applied to a stretched out version of the matrix. Any norm
on the vector space
R
mn
can be used as a matrix norm on
R
m
×
n
.
But matrices are also operators, and as such we can measure their size by how much
they stretch the vectors on which they operate:
k
A
k
D
,
R
= max
k
x
k
D
=1
k
Ax
k
R
.
You can think of it as the radius of the smallest ball centered at the origin of
R
m
that contains the image of the ball of radius 1 centered at the origin of
R
n
(the
unit
ball
in
R
n
). This is the norm
induced
by (or subordinate to) the vector norms
k · k
D
and
k · k
R
. For some authors, this is the
only
type of matrix norm.
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 Fall '10
 MARK
 Linear Algebra, Vector Space, Euclidean space, Hilbert space

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