3A.8 Quadratic Forms

Back

Consider a vector, x; and a square matrix, A: Then the function QðxÞ ¼ ðx; AxÞ is called a quadratic form.

For a real vector x and a real and symmetric matrix A;

QðxÞ ¼ xTAx

Positive definite matrix: If ðx; AxÞ . 0 for all x – 0; then A is said to be a positive definite matrix.

Also, the corresponding quadratic form is also said to be positive definite.

3-54 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Positive semidefinite matrix: If ðx; AxÞ $ 0 for all x – 0; then A is said to be a positive semidefinite

matrix. Note that in this case the quadratic form can assume a zero value for a nonzero x: The

corresponding quadratic form is also said to be positive semidefinite.

Negative definite matrix: If ðx; AxÞ , 0 for all x – 0; then A is said to be a negative definite matrix. The

corresponding quadratic form is also said to be negative definite.

Negative semidefinite matrix: If ðx; AxÞ # 0 for all x – 0; then A is said to be a negative semidefinite

matrix. Note that, in this case, the quadratic form can assume a zero value for a nonzero x: The

corresponding quadratic form is also said to be negative semidefinite.

Note: If A is positive definite, then 2A is negative definite. If A is positive semidefinite, then 2A is

negative semidefinite.

Principal minors: Consider the matrix

A ¼

a11 a12 · · · a1n

a21 a22 · · · a2n

.. .

an1 an2 · · · ann

2

66666664

3

77777775

Its principal minors are the determinants of the various matrices along the principal diagonal, as given by

D1 ¼ a11; D2 ¼ det

a11 a12

a21 a22

" #

; D3 ¼ det

a11 a12 a13

a21 a22 a23

a31 a32 a33

2

664

3

775

; and so on

Sylvester’s theorem: A matrix is positive if definite if all its principal minors are positive.

Навигация