Question

Sufficient conditions for a matrix to be positive definite

Answer 1

Answer:

Step-by-step explanation:

Positive semi-definite matrices are defined similarly, except that the above scalars {\displaystyle z^{\textsf {T}}Mz} {\displaystyle z^{\textsf {T}}Mz} or {\displaystyle z^{*}Mz} {\displaystyle z^{*}Mz} must be positive or zero (i.e. non-negative). Negative definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is called indefinite.

The matrix {\displaystyle M} M is positive definite if and only if the bilinear form {\displaystyle \langle z,w\rangle =z^{\textsf {T}}Mw} {\displaystyle \langle z,w\rangle =z^{\textsf {T}}Mw} is positive definite (and similarly for a positive definite sesquilinear form in the complex case). This is a coordinate realization of an inner product on a vector space.[2]