§

and . ¤

’w

¢¥ ¤A

¥

P#

1 1

§©

A nonsingular matrix is such that T

X

§ ¡

§

¦ ¦(T

X

§ § ²t¢

for every vector if and only if is normal and .

¦ ¤

y

£ 6 A T

£

The suf¬cient condition is trivially true. To prove the necessary condition, assume ¢ ¢

X

¢¦ © §

¬ § m¢

²

that, for any vector , where is a polynomial of degree . Then ¦ ¦ © y

§ §

it is easily seen that any eigenvector of is also an eigenvector of . Therefore, from

§ §

Theorem 1.2, is normal. Let be the degree of the minimal polynomial for . Then,

§ ²

¢

since has distinct eigenvalues, there is a polynomial of degree such that

y

¨ ¡¡µ£ „ ¢

§|5¥ yq¢| ¢ £ ¥§

„ 5| j C¦£¥

5§

¡ C

"–© ¡"

§

1B

$

T

XR T T

¬ ©§

¬ l

¬

¢ ¢

R©

for . According to the above argument, for this , it holds

X© V( E

( X

T

y¤

§ §

²

¢ and therefore . Now it must be shown that . Let be a (nonzero) ¢ ¥

X

T ©

§ ¡ ¥ §

y

vector whose grade is . By assumption, . On the other hand, we also ¥(

X

©

§ §¢¬ ¥ W “ W( ( §W(

§ ¥ § l

²

have . Since the vectors are linearly independent,

¥ ¥ ¥ © y

m(¢ (

² W§ §

must not exceed . Otherwise, two different expressions for with respect to the ¥ ©

¥ W y“

’§ ¬¥ § §

basis would result and this would imply that . Since is

¥© ( ¥ Y

¬

nonsingular, then , which is a contradiction. ¥ Y

T

Proposition 6.14 gives a suf¬cient condition for DIOM(s) to be equivalent to FOM.

X

§ §

According to Lemma 6.3, this condition is equivalent to being normal and ¤

T

t¢

² . Now consider the reverse result. Faber and Manteuffel de¬ne CG(s) to be the class XR u

T

y § ¬

of all matrices such that for every , it is true that for all such that ¦ ¦( ©

¦ Y V™E

(

² XW¦ W

. The inner product can be different from the canonical Euclidean

¢

wE

y

dot product. With this de¬nition it is possible to show the following theorem [85] which is

stated without proof.

T T T

cv ˜¤

¥¦ ¥¡ ¢ X

X X

¤A

¥

' 1

¡§ § §T § µ¢

²

, if and only if or is normal and .

¢

¦) ¤

y

X

§

It is interesting to consider the particular case where , which is the case of ¤

y §

the Conjugate Gradient method. In fact, it is easy to show that in this case either has a

minimal degree , or is Hermitian, or is of the form T

y R X

¡

¬§ w ¢

¡

©

¢

²¬

where and are real and is skew-Hermitian, i.e., . Thus, the cases in ¢ ¢ ¢

which DIOM simpli¬es into an (optimal) algorithm de¬ned from a three-term recurrence

are already known. The ¬rst is the Conjugate Gradient method. The second is a version

of the CG algorithm for skew-Hermitian matrices which can be derived from the Lanczos

algorithm in the same way as CG. This algorithm will be seen in Chapter 9.

—E“fz g cz•tc™¡ l

— ™™

9£¡

£

The convergence behavior of the different algorithms seen in this chapter can be analyzed

by exploiting optimality properties whenever such properties exist. This is the case for

the Conjugate Gradient and the GMRES algorithms. On the other hand, the non-optimal