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Page 11-8
Singular value decomposition
To understand the operation of Function SNRM, we need to introduce the
concept of matrix decomposition. Basically, matrix decomposition involves
the determination of two or more matrices that, when multiplied in a certain
order (and, perhaps, with some matrix inversion or transposition thrown in),
produce the original matrix. The Singular Value Decomposition (SVD) is such
that a rectangular matrix
A
m
×
n
is written as
A
m
×
n
=
U
m
×
m
⋅ S
m
×
n
⋅ V T
n
×
n
,
Where
U
and V are orthogonal matrices, and
S
is a diagonal matrix. The
diagonal elements of
S
are called the singular values of A and are usually
ordered so that s
i
≥
s
i+1
, for i = 1, 2, …, n-1. The columns [
u
j
] of
U
and [
v
j
] of
V
are the corresponding singular vectors. (Orthogonal matrices are such that
U ⋅ U T
=
I
. A diagonal matrix has non-zero elements only along its main
diagonal).
The rank of a matrix can be determined from its SVD by counting the number
of non-singular values. Examples of SVD will be presented in a subsequent
section.
Functions RNRM and CNRM
Function RNRM returns the Row NoRM of a matrix, while function CNRM
returns the Column NoRM of a matrix. Examples,
Row norm and column norm of a matrix
The row norm of a matrix is calculated by taking the sums of the absolute
values of all elements in each row, and then, selecting the maximum of these
sums. The column norm of a matrix is calculated by taking the sums of the
absolute values of all elements in each column, and then, selecting the
maximum of these sums.