User Manual

86 Section 4: Using Matrix Operations
86
 
ASAA  min)(1 K
and
 
SAA
1


min1
,
where the minimum is taken over all singular matrices S. That is, if K(A) is large, then the
relative difference between A and the closest singular matrix S is small. If the norm of A
−1
is
large, the difference between A and the closest singular matrix S is small.
For example, let







9999999999.1
11
A
Then










1010
10
1010
10999,999,999,9
1
A
and ||A
−1
|| = 2 × 10
10
. Therefore, there should exist a perturbation ΔA with ||ΔA|| = 5 ×10
−11
that makes A + ΔA singular. Indeed, if











11
11
1050
1050
ΔA
with ||ΔA|| = 5 ×10
−11
, then







59999999999.1
59999999999.1
ΔAA
and A + ΔA is singular.
The figures below illustrate these ideas. In each figure matrix A and matrix S are shown
relative to the "surface" of singular matrices and within the space of all matrices. Distance is
measured using the norm. Around every matrix A is a region of matrices that are practically
indistinguishable from A (for example, those within rounding errors of A). The radius of this
region is ||ΔA||. The distance from a nonsingular matrix A to the nearest singular matrix S is
1/||A
−1
||.