B Q 0 RS!

refers to the algebraic multiplicity of the individual eigenvalue and is the index of the

B @4

) )

T! @ 3

R

@3

eigenvalue, i.e., the smallest integer for which .

C C

§CjG b¤

¥¦ ¥¡ U

Q T

Any matrix can be reduced to a block diagonal matrix consisting of

diagonal blocks, each associated with a distinct eigenvalue . Each of these diagonal @

¡

blocks has itself a block diagonal structure consisting of sub-blocks, where is the § §

geometric multiplicity of the eigenvalue . Each of the sub-blocks, referred to as a Jordan

@

@

u d

u ¡

¡£ ¢ ¡ ¥ ¥ £ £ ¥ ¡ £ $ ¤¢ ¥ £ ¥ ¥ ¢

£¥

§

©

# Q

block, is an upper bidiagonal matrix of size not exceeding , with the constant @

on the diagonal and the constant one on the super diagonal.

111)‘ X

The -th diagonal block, , is known as the -th Jordan submatrix (sometimes

Y — 6 — 0 ‘

¡

—

“Jordan Box”). The Jordan submatrix number starts in column §§¥

¥¥

—0 Y

. Thus,

&

344 @8

99

0

44 99

6$

44 99

..

.

¡ 0 0 9

4

Y

0 &

..

5 A

.

#

where each is associated with and is of size the algebraic multiplicity of . It has

@ @

itself the following structure,

344

9@8

9 344 @8

99

0 @

6 .. ..

' A

. .

with

..

5

A 5

. @

R £Y

¢ @

'

Each of the blocks corresponds to a different eigenvector associated with the eigenvalue

Q

. Its size is the index of .

@ @

§¨ ¥¤¥B£¢

¦ % ¡¡ ¥!5342P#!IG'8##I

§%

6 § 1 '

Here, it will be shown that any matrix is unitarily similar to an upper triangular matrix. The

only result needed to prove the following theorem is that any vector of 2-norm one can be

˜ ‘

3

completed by additional vectors to form an orthonormal basis of .

SG b¡¤

§¥¦ ¥ Q ©T 0

For any square matrix , there exists a unitary matrix such that

0I0 5

is upper triangular.

£ ˜

©¨¦

§ §¥ T

˜

The proof is by induction over the dimension . The result is trivial for .

˜ ˜

3

Assume that it is true for and consider any matrix of size . The matrix admits

at least one eigenvector that is associated with an eigenvalue . Also assume without

V @

(6 y B

loss of generality that . First, complete the vector into an orthonormal set, i.e.,

V V

© ¨

V1 3

˜¨ s˜

s˜

˜ 3

¨

¬nd an matrix such that the matrix is unitary. Then

C 3 1 ©

and hence, 6@

V

¨ I t

3 1 I I ¨ I ©

© V @ V ¨ £ & §¥