Definite semi definite definite semi definite functions Definite

+ Definite, + semi definite, - definite & - semi definite functions

+/- Definite quadratic function •

+/- Definite quadratic function … •

Indefinite quadratic function •

+/- Definite matrix • => Matrix P is converted into quadratic function. So P is + definite if its quadratic function is + definite

+/- Definite matrix … •

Test for + definite matrix (Sylvester’s criterion) • Sylvester’s criterion is applicable for symmetrical matrix only • For a symmetric matrix P to be + definite matrix(P>0) 1. All diagonal elements must be +ve and non zeros(>0). 2. All the leading principal minors (determinants) must be +ve and non zeros (>0).

Examples • Half-Half (To make P symmetrical ) P is symmetrical matrix

Example 1… •

Example 1… • Both conditions are satisfied so P is + definite and hence f(x) is + definite quadratic function.

Test for + semi definite matrix (Sylvester’s criterion) • For a symmetric matrix P to be + semi definite matrix (P≥ 0) 1. All diagonal elements must be +ve and some may be zeros (≥ 0). 2. All the principal minors (determinants) must be +ve and some may be zeros (≥ 0). (Note: Principal minors not leading principal minors as in + definite matrix )

Principal minors of a matrix •

Example •

Example… •

Test for - definite matrix (Sylvester’s criterion) • Sylvester’s criterion is applicable for symmetrical matrix only • For a symmetric matrix Q to be - definite matrix (Q=-P) 1. All diagonal elements must be -ve and non zeros(<0). 2. All the leading principal minors (determinants) with even order must be +ve and non zero(>0). 3. All the leading principal minors (determinants) with odd order must be -ve and non zero(<0). Note: 1. Alternate sign: 1 st order = -ve, 2 nd order = +ve, 3 rd order = -ve, …. . 2. If matrix P is + definite then –P will always be –definite matrix.

Test for - definite matrix (Sylvester’s criterion)… • Note: All the elements of + definite matrix P are multiplied by –ve sign

Example •

Test for – semi definite matrix (Sylvester’s criterion) • Similar to relation between + definite and – definite matrix • Sylvester’s criterion is applicable for symmetrical matrix only • For a symmetric matrix Q to be – semi definite matrix (Q=-P) 1. All diagonal elements must be -ve and some may be zeros(≤ 0). 2. All the principal minors (determinants) with even order must be +ve and some may be zero (≥ 0). 3. All the principal minors (determinants) with odd order must be -ve and some may be zero (≤ 0). Note: 1. Alternate sign: 1 st order = -ve, 2 nd order = +ve, 3 rd order = -ve, …. . 2. If matrix P is + semi definite then –P will always be – semi definite matrix. 3. Proof can be done as we did in case of – definite matrix from + definite matrix.

Example •

Next: • Now some other concepts like Optimality conditions.