Introduction to Computational Linear Algebra

Gauss Elimination and LU Decompositions of Matrices 67

1. Symmetric Matrices having pmp. In such case, one has A = A

T

.

The operations can be reduced by half. See Exercises 3.8, 3.9 and Computer

Exercise 3.1.

2. Symmetric Positive Definite

Definition 3.6 A matrix A ∈ R

n×n

is said to be positive definite if:

x

T

Ax > 0, ∀x ∈ R

n

, x 6= 0, (x

T

Ax = 0, if and only if x = 0).

An immediate consequence of this definition is the following proposition.

Proposition 3.3 A positive definite matrix has the principal minor property

(pmp).

Proof. Select x ∈ R

n×n

, such that x 6= 0 and x

k

= 0, k > i, 1 < i ≤ n. Then:

0 < x

T

Ax = (x

(i)

)

T

A

i

x

(i)

where A

i

is the i

th

principal minor of the ...