HP 15c - Manual (Page 86)

86 Section 4: Using Matrix Operations
86
 
A S A A
 
min ) ( 1 K
and
 
S A A
1   
min 1 ,
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
1 1
A
Then








 
10 10
10
10 10
10 999 , 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
10 5 0
10 5 0
ΔA
with ||ΔA|| = 5 ×10
−11
, then






 
5 9999999999 . 1
5 9999999999 . 1
ΔA
A
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
||.