viewpoint is far more complex. Approximations may be available but it may be dif¬cult to

estimate how accurate they are. This clearly will depend on the data at hand, i.e., primarily

on the coef¬cient matrix. This section gives a very brief overview of the existence theory

as well as the sensitivity of the solutions.

¤ ¦ ¦

' 3 4II G ¨

¥ ¦¢

¡¡ ¨ ¨ !0 1

Consider the linear system

t

‚

¨ ¤ §¥

¦

¦

¦

Here, is termed the unknown and the right-hand side. When solving the linear system

(1.65), we distinguish three situations.

0 ‚

¦

Case 1 The matrix is nonsingular. There is a unique solution given by . &

B1 ¤ 1B B ¤

)

( )

( &¦

Case 2 The matrix is singular and . Since , there is an

B

) )

— C C

&¦ &¦

such that . Then is also a solution for any in . Since is

C C

at least one-dimensional, there are in¬nitely many solutions.

B¢ ¤

(

0

(

Case 3 The matrix is singular and . There are no solutions.

C

¥ U © §¦

T ¥

The simplest illustration of the above three cases is with small diagonal

matrices. Let

7

¢

7

¦

Then is nonsingular and there is a unique given by

p ‚ 7

¢

¦

Now let

7

7 7 7

1 ¤ P £ ¡

B )

(

Then is singular and, as is easily seen, . For example, a particular element

$&

C £

&¦ &¦ &¦

such that is . The null space of consists of all vectors whose ¬rst

&

&

¡

component is zero, i.e., all vectors of the form . Therefore, there are in¬nitely many ¡

Y h§ ¤¦¤¢ ¢ “¤¥

u u ©¢¥ £ u¥

£ ¡£ ¥U© ¡ ¡ ¥ © ¥ ¥

§¡ ©

¡

solution which are given by

p B

¢7 S

¦

C

Finally, let be the same as in the previous case, but de¬ne the right-hand side as

In this case there are no solutions because the second equation cannot be satis¬ed.

¤ ¦

2' G' !PG £ § §¥

¤¥ ¦¢

A¡ ¡ ¨

'

1 ¦

¡

j(˜

˜

Consider the linear system (1.65) where is an nonsingular matrix. Given any matrix

— CB #

£

¢ ¢ ¢ ¢

, the matrix is nonsingular for small enough, i.e., for where is

¢ ¢

some small number; see Exercise 32. Assume that we perturb the data in the above system,

1

B

¤

¢ ¢ £¦

¢