Introduction to Computational Linear Algebra

62 Introduction to Computational Linear Algebra

Consequently, let u

(k)

= α

(k)

∈ R

n

, such that

(α

(k)

)

T

= (0, ..., 0, α

(k)

k+1

, ..., α

(k)

n

), where α

(k)

i

=

x

i

x

k

, i = k + 1, ..., n.

A Gauss transform can then be defined as follows:

Definition 3.4 Let M

(k)

∈ R

n×n

be such that

M

(k)

= I − α

(k)

e

T

k

,

α

(k)

is said to be the Gauss vector with its components α

(k)

i

, i = k + 1, ..., n

being the “multiplying factors” of the k

th

level Gauss transform.

Note that M

(k)

is a lower triangular matrix of the form:

M

(k)

=

















1 ... 0 0 ... 0

. ... . . ... .

0 ... 1 0 ... 0

0 ... −α

(k)

k+1

1 ... 0

. ... . . ... .

0 ... −α

(k)

n

0 ... 1

















Thus, from Proposition 3.2 one has:

M

(k)

x =





















x