University Mathematics Handbook (2015)

X. Algebra

Chapter 12. Quadratic Form

a. The expression , when is a symmetric matrix in and is called quadratic form.

b. Example in :

A quadratic form is positive (non-negative) if for every

,

c. A quadratic form is negative (non-positive) if for every , .

d. The following propositions are equivalent

1.  Quadratic form is positive.

2.  The eigenvalues of are positive.

3.  All numbers

,…..,detA (*)

are positive.

e. Quadratic form is negative if and only if the signs of the n numbers in (*) change intermittently, the first being negative.