Why do positive definite matrices have to be symmetric? [duplicate] This question already has an answer here:
Non-Symmetric Positive Definite Matrices
3 answers
Definitions of positive definiteness usually look like this: A symmetric matrix M is positive definite if xT M x 0 for all vectors x neq 0. Why must M be symmetric The definition seems to make sense for general square matrices. Let quadratic form f be defined by f mathrm x : mathrm xtop mathrm A ,mathrm x where mathrm A in mathbbRn times n. Since mathrm xtop mathrm A ,mathrm x is a scalar, then mathrm xtop mathrm A ,mathrm xtop mathrm xtop mathrm A ,mathrm x, i.e., mathrm xtop mathrm Atop mathrm x mathrm xtop mathrm A ,mathrm x. Hence, mathrm xtop leftfracmathrm A - mathrm Atop2right mathrm x 0 Thus, the skew-symmetric part of matrix mathrm A does not contribute anything to the quadratic form. What is left is, then, the s...