—™ —‘–

¢£¡

In this section we consider projection methods on Krylov subspaces, i.e., subspaces of the

form T

r±0 „ ®

i

£

X

§ ¢

W§(¦ W§¤©’§¦ §

¢¡

! H

C

¦( (¦ ( © ¦W “

(

¢

which will be denoted simply by if there is no ambiguity. The dimension of the sub- ¢

space of approximants increases by one at each step of the approximation process. A few

elementary properties of Krylov subspaces can be established, many of which need no

T

«ª

proof. A ¬rst property is that is the subspace of all vectors in which can be writ- ¢

X

§ T (©

¬ ²

ten as , where is a polynomial of degree not exceeding . Recall that

¦ !

¡ ¡

y

the minimal polynomial of a vector is the nonzero monic polynomial of lowest degree ¦ ¡

¬ ¦X § §

such that . The degree of the minimal polynomial of with respect to is often ¦

Y

¡

§

called the grade of with respect to , or simply the grade of if there is no ambiguity.

¦ ¦

¤

A consequence of the Cayley-Hamilton theorem is that the grade of does not exceed . ¦

The following proposition is easy to prove.

´ tC

| ¢ £ j§

¨

9

"–© ¡§’

¥

Q¤ ¡ v A£

£¦ DA

C

' ' ©' 2 )

§)

¬

§

Let be the grade of . Then is invariant under and

¡

¦

¢

for all .

¢

!

It was mentioned above that the dimension of is nondecreasing. In fact, the fol- ¢

lowing proposition determines the dimension of in general. ¢

Q¤ ¡ v A£

£¦ £¤A

' ' ©' 2 )

§)

The Krylov subspace is of dimension if and only if the

¢ !

§

grade of with respect to is not less than , i.e.,

¦

T T

!

¬X X

¥

¡

£ H ¥¦

¤

54

6 !

¦

¢ !

Therefore, T T

X X

¬ 564 ¥

¡

H ¤¦¦ ( "§ 5I6¥

! C4 © ¦

¢

6£

A

£

¢

¦ W “ W¤©W(¤¦

§( ( ¦ §

The vectors form a basis of if and only if for any set of ¢

!#(¡Y teE( R R R “ R !

² (¬ R

scalars , where at least one is nonzero, the linear combination

£ ¢

yT ¦ § W

is nonzero. This is equivalent to the condition that the only polynomial of

X

²!

Y¬¦§

degree for which is the zero polynomial. The second part of the ¡

y

proposition is a consequence of the previous proposition.

Q¤ ¡ v A£

£¦ ¥¤A

' ' ©' 2 )

§)

§

Let be any projector onto and let be the section of

¤ ¢ ¢ ¢

² §! ¬§ §¨§ ¢ ¤ ¢

to , that is, . Then for any polynomial of degree not exceeding

¢ ¢

¤

, T T

y X X

§ ¢ ¦ §

¬

¢ ¦

(

¢

and for any polynomial of degree ,

T! T

§ ¢ ¬ ¦ X §T ¢ ¤ T X

¦

¢ ¢

£ ²!

§ ¢ ¬ ¦X § ¢

X

6 A ¢ XT

£

First we prove that for any polynomial of degree . It ¦ ¢ RT

y ²##(V`7(