'#

is nonsingular.

(

0

¨ 7

.

P

&

In reality, the four conditions in the above de¬nition are somewhat redundant and

equivalent conditions that are more rigorous will be given later. Let be any matrix which

satis¬es properties (1) and (2) in the above de¬nition and let be the diagonal of . Since 7

7 7

,

£C 0

¢7 C

7B 3 7 73B3

¡

3 &

Now de¬ne

0

˜ 73 &

(

0Y

˜ 0 C˜ 0

3 3

B 7

Using the previous theorem, is nonsingular and 7 7I

& & &

C ˜ B ¡

if and only if . It is now easy to see that conditions (3) and (4) of De¬nition 1.4

C ˜B ¡

can be replaced by the condition .

„ SG b¡¤

§¥¦ ¥

Q T

Let a matrix be given such that

„

˜}ia)`1x1Y # „‚‚

Yy

" 7 for .

#

˜ i)11Y

"… 7 for .

Then is an -matrix if and only if

0 7 ˜ C ˜ ©¡ 3

'#

B , where .

&

©¨¦£

§ §¥

T

From the above argument, an immediate application of Theorem 1.15 shows that

˜ B0 ¡ – – ˜

3

properties (3) and (4) of the above de¬nition are equivalent to , where

0 7 – C

–

and . In addition, is nonsingular iff is and is nonnegative iff is.

& &

The next theorem shows that the condition (1) in De¬nition 1.4 is implied by the other

three.

¡ „ SG b¡¤

§¥¦ ¥

Q T

Let a matrix be given such that

˜ i)11Y}a`xY # „‚

"… 7

for .

is nonsingular.

( 0

'#

7

.

&P

Then

0 ˜ i)1 1Yy# ©‚¡ "

¨ 7 for , i.e., is an -matrix.

3

'©

˜B

˜

7 where .

&

C

uY u d¥ £ ¡ ¨ ¢¤¥ ¤¤§ ¥ §¢

n ¦¢¥ £ ¡© ¡

© ¥ ¡ © ¥ ¡ ¥ © ¥ ¥

¥ ¨

£ y V C ( B 0

§ § ¥¦ T

–

–

De¬ne . Writing that yields

I

&

Y

' ' ‚ 1‘)(&'

0

which gives

' ' ‚ ‘ & q q ‚

3

¡£

££

R ¢

( ( V

' ' ‚ V ‚

7 7 7

Since for all , the right-hand side is and since , then .

#

The second part of the result now follows immediately from an application of the previous

theorem.

Finally, this useful result follows.

˜ „ §CjG b¤

¥¦ ¥¡

Q U

T

Let be two matrices which satisfy

s# # H…

˜ .

!

7 for all .

˜

Then if is an -matrix, so is the matrix .