HP 15c - Manual (Page 84)

84 Section 4: Using Matrix Operations
84
ILL-Conditioned Matrices and the Condition Number
In order to discuss errors in matrix calculations, it's useful to define a measure of distance
between two matrices. One measure of the distance between matrices A and B is the norm of
their difference, denoted ||A − B||. The norm can also be used to define the condition number
of a matrix, which indicates how the relative error of a calculation compares to the relative
error of the matrix itself.
The HP-15C provides three norms. The Frobenius norm of a matrix A, denoted ||A||
F
, is the
square root of the sum of the squares of the matrix elements. This is the matrix analog of the
Euclidean length of a vector.
Another norm provided by the HP-15C is the row norm. The row norm of an m

n matrix A
is the largest row sum of absolute values and is denoted ||A||
R
:


 

n
j
ij R
a
m i
1
1
max
A
The column norm of the matrix is denoted ||A||
C
and can be computed by ||A||
C
= ||A
T
||
R
. The
column norm is the largest column sum of absolute values.
For example, consider the matrices














6 5 4
2 2 2
and
9 5 4
3 2 1
B A
Then






 
3 0 0
1 0 1
B A
and
||A + B||
F
= 3 . 3 11

(Frobenius norm),
||A − B||
R
= 3 (row norm), and
||A − B||
C
= 4 (column norm).
The remainder of this discussion assumes that the row norm is used. Similar results are
obtained if any of the other norms is used instead.
The condition number of a square matrix A is defined as
K(A) = ||A|| ||A
−1
||.
Then 1 ≤ K(A) < ∞ using any norm. The condition number is useful for measuring errors in
calculations. A matrix is said to be ill-conditioned if K(A) is very large.
If rounding or other errors are present in matrix elements, these errors will propagate through
subsequent matrix calculations. They can be magnified significantly. For example, suppose