1) 26(

1)

3

C ˜ B C ˜ B … 0

3 3

where is an matrix obtained by deleting the ¬rst row and the -th

C B 20(

1)

8

7

column of . A matrix is said to be singular when and nonsingular otherwise.

We have the following simple properties:

• }˜ B B C ( BB

90(

1) 190(

)

˜

.

• C

90(

1) 90(

1)

C G B

.

• CB

90(

1) 9)0(

1

B ‘ C 1B .

• C

90(

1) 20(

1)

C Q B

.

• C

C

90(

1)

.

From the above de¬nition of determinants it can be shown by induction that the func-

B C B 126(

)

@3

tion that maps a given complex value to the value is a polynomial @ AB @

¡

C

˜

of degree ; see Exercise 8. This is known as the characteristic polynomial of the matrix

.

¥ „ ¤q

SRP I#GDE C

H QF F HF U

T

A complex scalar is called an eigenvalue of the square matrix if @

t

‘

a nonzero vector of exists such that . The vector is called an eigenvector

V W

V VU@ V

of associated with . The set of all the eigenvalues of is called the spectrum of and

@

B

is denoted by . X

C

C

20(

1)

B BB

@3 a

7

A scalar is an eigenvalue of if and only if . That is true

@ A @

¡

C `

Y

if and only if (iff thereafter) is a root of the characteristic polynomial. In particular, there

@

˜

are at most distinct eigenvalues.

It is clear that a matrix is singular if and only if it admits zero as an eigenvalue. A well

known result in linear algebra is stated in the following proposition.

V¤ G S£

£¦ „

SRI FcP` `

Q

bQ

H QF U

T

A matrix is nonsingular if and only if it admits an inverse.

‘u @¥ £ ¡ ©§¢ ¥ ¥

u n ¨¦

¤¤§ ¢¢¢

¥¡

£ ¡© ¥ ¡ © ¥ ¡ ¥ © ¥ ¥

¥ ¨

Thus, the determinant of a matrix determines whether or not the matrix admits an inverse.

The maximum modulus of the eigenvalues is called spectral radius and is denoted by

©¡

B

§¦¥©¤ ¢£ C ©B¡

C

¨ @

A

The trace of a matrix is equal to the sum of all its diagonal elements

q ‚ 1‘ ( & C B

1

0

It can be easily shown that the trace of is also equal to the sum of the eigenvalues of

counted with their multiplicities as roots of the characteristic polynomial.

q¤ G S£

£¦

U SRFP cbP` Q`

HQ F Q

T Q@

If is an eigenvalue of , then is an eigenvalue of . An IH

@

Q@

eigenvector of associated with the eigenvalue is called a left eigenvector of .

H

I

When a distinction is necessary, an eigenvector of is often called a right eigenvector.

Therefore, the eigenvalue as well as the right and left eigenvectors, and , satisfy the

@ V

relations

I I