Introduction to Computational Linear Algebra

Basic Concepts for Matrix Computations 37

with V

(1)

being the unitary matrix representing the base {f

1

, f

1

2

, ..., f

1

n

}. As

A

(1)

f

1

= λ

1

f

1

, then the first column of the matrix A

(1)

is (λ

1

0 .... 0)

∗

and

A

(1)

has the following form:

A

(1)

=

λ

1

A

(1)

12

O

n−1,1

A

(1)

22

!

where O

n−1,1

is a zero column vector of size n − 1, A

(1)

12

∈ C

1×(n−1)

and

A

(1)

22

∈ C

(n−1)×(n−1)

.

From (2.8), one has A = V

(1)

A

(1)

(V

(1)

)

∗

and A−λI = V

(1)

(A

(1)

−λI)(V

(1)

)

∗

.

So that

p

A

(λ) = p

A

(1)

(λ),

and therefore, Λ(A

(1)

) = Λ(A).

We may then start the recurrence process and consider the eigenvalues of A

(1)

22

,

given that

p

A

(1)

(λ) = (λ − λ

1

)p

A

(1)

22

(λ),

and therefore,

Λ(A

(1)

22

) ⊆ Λ(A),

with Λ(A

(1)

22

) = Λ(A) − {λ

1

} if and ...