¨ VU@

DI ¨

¨ 7

˜B C ˜B

C ¨(I ¨ 0 ˜

3 3

Now use the induction hypothesis for the matrix : There

! ˜B C ˜B

C

0 0 ˜ 0I 0 00 3 3

exists an unitary matrix such that is upper triangular.

5

uY u d¥ £ ¡ ¨ ¢¤¥

n ¦¢¥

¤¤§ ¥ §¢

¡

£ ¡© ¥ ¡ © ¥ ¡ ¥ © ¥ ¥

¥ ¨

‚!˜

˜

De¬ne the matrix

0 (0 7

0 7 0

0 0

( ( 0

and multiply both members of (1.24) by from the left and from the right. The 0 IP

resulting matrix is clearly upper triangular and this shows that the result is true for , with

© 0 (0

0 h!˜

˜

which is a unitary matrix.

A simpler proof that uses the Jordan canonical form and the QR decomposition is the sub-

ject of Exercise 7. Since the matrix is triangular and similar to , its diagonal elements

5

are equal to the eigenvalues of ordered in a certain manner. In fact, it is easy to extend

the proof of the theorem to show that this factorization can be obtained with any order for

the eigenvalues. Despite its simplicity, the above theorem has far-reaching consequences,

some of which will be examined in the next section.

0 ˜ #

It is important to note that for any , the subspace spanned by the ¬rst columns

#

0 5¤0

of is invariant under . Indeed, the relation implies that for , we #

have

… ( &

•) p… ' 01( (`…

) ¢

)

11)1 6 0 1) ' 0 3'

0

) ) '

If we let and if is the principal leading submatrix of dimension 5

of , the above relation can be rewritten as

5

' 5' 0 ' 0

which is known as the partial Schur decomposition of . The simplest case of this decom-

) )

0

position is when , in which case is an eigenvector. The vectors are usually called

Schur vectors. Schur vectors are not unique and depend, in particular, on the order chosen

for the eigenvalues.

A slight variation on the Schur canonical form is the quasi-Schur form, also called the

real Schur form. Here, diagonal blocks of size are allowed in the upper triangular

matrix . The reason for this is to avoid complex arithmetic when the original matrix is

5

real. A block is associated with each complex conjugate pair of eigenvalues of the

matrix.

¥

U

T

¡w

§¦

¥ ©

Consider the matrix

7 3 87

A

3

5 7

3

The matrix has the pair of complex conjugate eigenvalues

1) Yup ¡ 11–¤¥¤£ ¢ z

7 7

¨

@

u d

u

¡£ ¢ ¡ ¥ ¥ £ £ ¥ ¡ £ $ ¤¢ ¥ £ ¥ ¥ ¢

£¥

§

© ¡

1) £ qz

¡7

and the real eigenvalue . The standard (complex) Schur form is given by the pair

of matrices

Y£ z 1A£ V£ p £ Y p 3 Y£ z¢ z z ¥ p 3

z p

z

8 ¥ ¢ qz 7 3 7 ¥¢£ 7 73 ¤7 7

¤ ¤

p ¥ ¤£ p 3 qz ¢ Y V p 5 ¨

A ¢ z 7 3 ¢7 7 7 W3

7 73 7W3

¤

£ z p

—

¦7 ¢¡ 7 7W3 3 ¢7 ¤7

¢ £¤

¤ ¥