respect to the zeros. Also, they assume that there are many zero elements or that the matrix

is of low rank. This is in contrast with the classi¬cations listed earlier, such as symmetry

or normality.

C

‘ and

Euclidean inner product. The Euclidean inner product of two vectors

¦ ¦

"$$$

%%%%" 01( g %‚ B

£

, a “canonical” inner product is the

In the particular case of the vector space ‘

which yields the result (1.2).

C

¤

¨

¨ ¤

© ¦

¦

3

¤

© ¦

¨

C

C ™xB B C x B

¤ ¤

© ¨

¨ ©"

¨¨ ¤ ¤

)

C ™B ¦

© ¦ ¨ ¨ "

¦

3 @ ¦©

3¦ @3 #7

¤ ¤

— C6 C ™ B C C ™ B

¦

¨ "

¨¦

l x B C

6 g B 6 g B

¤ ¤ ¤

shows the above equality

. Then ¨ ¦

¨ ¦ ¨ "

¨ ¨

¦

7 @3 @3

(

yB C

( C y B

@ and take then the inequality is trivially satis¬ed. Assume that If

¨ ¨

7 7

gB

¤ ¤ ¤ ¤ ¤

@ $@ '@

©

¨¨ © ¨

¦

¨¦ ¦

¦ ¨ ©¨

¦ ¦

3 3 @3 @3 B

C ™ B 6 — C ™ B C g B Q C g B C ™

¤

,

C

¤

using the properties of The proof of this inequality begins by expanding

¨ ¦ ¨

@ 3P© @ P¦

3

™B

¤ ¤

% ¤

¨© ¨ ©"

¦¦ ¨©

¦

¨ & ¦§¥

t C ™ B C g B # 6 C x B

Cauchy-Schwartz inequality:

as can be readily shown. A useful relation satis¬ed by any inner product is the so-called

¤ ¤

iff and

¦ "

¦¦ ¦¦

©

a

7 a

7 7

z‚

C g B ( C x B

the condition (3) can be rewritten as

¤ ¤ ¤

for any and . In particular for any . Hence,

Similarly,

¨ ¦ "

¦ ©

¨ ¨ ¨©

8

7 7 7 7 #

7

C g B C z B C pB

¤ ¤ ¤

¨© ¦ ¨ ¦ ¦

a

7 78 7 7

z C x B p C z g B C g B

must also be positive for any nonzero . For any and , ¨ ¦ ¦

C C

¤ ¤

is real and therefore, (3) adds the constraint that

Note that (2) implies that

©"

¦¦ ¦¦

©

gB

xB

C

¤

¦ ¦

¦

7 7

z

p

gB C

¤

is positive de¬nite, i.e., ¨© ¦

$#

xB

§

£ ¤ ¤

©

¨¦ "

¨¦

¦¨

x C g B C ™ B C