„ § ¡|

¢

£ £

¢

˜h¤ v ¢

´¡¦ ¥ 7¦ §5 PD £S©E©U¥V8

£4CDA

¢3 ¢

5 £¡9US DV8

¢ S D 5¤D

6Q

DS

0 (%&$

'#

) 2

1

1. Until convergence, Do:

& ¦ ( ( ¦ %¬

2. Select a pair of subspaces and ¥

¬

3. Choose bases and for and

¥% & ¢ ¥ (V(

$ $

¢

3

©T

W W

X V§ ²A¨ ¬ ¨b

4. 3 ¥ ¥

§ w © ¬¬ ©

5. ¨

W“

3 b

6.

7. EndDo

¥

§

The approximate solution is de¬ned only when the matrix is nonsingular,

§

which is not guaranteed to be true even when is nonsingular.

¥ ´

7 §¨ ¦ ¥£

¤¢

£¤A

As an example, consider the matrix

¬§ (

¥

¥ ¥ "

¬ ¬

where is the

identity matrix and is the zero matrix, and let

! ! ! !

¥

§ §

& ¢ ¡ (( £ ¡ (

. Although is nonsingular, the matrix is precisely the block in

$¡

W §

the upper-left corner of and is therefore singular.

¥

§

There are two important particular cases where the nonsingularity of is guar-

anteed. These are discussed in the following proposition.

Q¤ ¡ v A£

£¦

´ DA

C

' ' ©' 2 )

§)

§

Let , , and satisfy either one of the two following conditions,

§ ¦¦ ¦

¬

is positive de¬nite and , or

§¦

¦

§ ¬

§ is nonsingular and .

¥ ¥

¬ §

Then the matrix is nonsingular for any bases and of and , respec-

¢

tively.

£ 6 A ¥

£

¦

Consider ¬rst the case (i). Let be any basis of and be any basis of . In

¥ ¥

¥ ¬

fact, since and are the same, can always be expressed as , where is a

¦

nonsingular matrix. Then ! !

¥

¬ "§

¬ §

¢ ¦4

§ §

Since is positive de¬nite, so is , see Chapter 1, and this shows that is non- ¢

singular.

¥

¬

Consider now case (ii). Let be any basis of and be any basis of . Since

¥ ¥

§ ¥

¬ §

, can be expressed in this case as , where is a nonsingular ¦ ¦

matrix. Then T

! !

a± q— ®

i

X

¥

"¢

¬ ¬§ § §

4

¦ ¦

T

X

§ ! ¥¤ § § §

Since is nonsingular, the matrix is of full rank and as a result, is