T

Assume that is an -matrix and let denote the diagonal of a matrix .

0

¥7

¤

The matrix is positive because

7 £

( p

A7 7

7 £

˜0

˜s

73

Consider now the matrix . Since , then

#

&£

(0 (˜ ¡

3A 3

7 7 £

which, upon multiplying through by , yields

7

&A

( 0 7( ˜ ¡( ˜

0 0 0

C˜

7 B &A 7 7 B &£

73 3 3 73

&A &£

£ £

C

0 0

˜

73 73

Since the matrices and are nonnegative, Theorems 1.14 and 1.16

&£ &A

imply that

¡ ¡ C ˜0 ‘Q C 0

7 3 B© 7 3 B© #

&£ &A

This establishes the result by using Theorem 1.16 once again.

$

) ©¨ £ ¤( § % £( £ ¢ ! @ %¤ £( ) "

¥ £

§

b1{˜{

{|

A real matrix is said to be positive de¬nite or positive real if

t

B z

p j

‘ ¨ ¦ £ §¥

¦

7 7

¤V

V V V

C

d „ PnA ”§ p£

u u © u u ud

£ £ ¡£ ¢ ¡ ¥ ¥ § ¥ ¤©

£

©

§

©

§ ©

It must be emphasized that this de¬nition is only useful when formulated entirely for real

B

variables. Indeed, if were not restricted to be real, then assuming that is real

V W

V V

C

for all complex would imply that is Hermitian; see Exercise 15. If, in addition to

V

De¬nition 1.41, is symmetric (real), then is said to be Symmetric Positive De¬nite

(SPD). Similarly, if is Hermitian, then is said to be Hermitian Positive De¬nite (HPD).

Some properties of HPD matrices were seen in Section 1.9, in particular with regards

to their eigenvalues. Now the more general case where is non-Hermitian and positive

de¬nite is considered.

We begin with the observation that any square matrix (real or complex) can be decom-

posed as

t

— ¡ ¨ & £ §¥

¦

in which

t

¨ ¢ £ ¦§¥

I ™

—B

C

t

C I B A ¨ £ £ ¦§¥

3

Note that both and are Hermitian while the matrix in the decomposition (1.42)

is skew-Hermitian. The matrix in the decomposition is called the Hermitian part of

£ £

, while the matrix is the skew-Hermitian part of . The above decomposition is the

¨ £ — £ £ ¦

analogue of the decomposition of a complex number into ,

B ¤¥G Q £ — £ B £ B7 ¢£‚

C Q B 3 C C C 3

¤

¦ ¨

B

When is real and is a real vector then is real and, as a result, the decom-

V W

V V

C

position (1.42) immediately gives the equality

t

C ¦ B% C B ¨ ¤ £ §¥

¦

V V V V

This results in the following theorem.

„ SG b¡¤

§¥¦ ¥ Q ¡T

Let be a real positive de¬nite matrix. Then is nonsingular. In

7

addition, there exists a scalar such that

( C B 66 t

¨ £ §¥

¦

V V V

for any real vector . V