Why do positive definite matrices have to be symmetric? [duplicate]

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 symmetric part fracmathrm A mathrm Atop2 which is diagonalizable and has real eigenvalues and orthogonal eigenvectors, all nice properties. Addendum Taking affine combinations of mathrm A and mathrm Atop, mathrm xtop gamma mathrm A 1-gamma mathrm Atop mathrm x f mathrm x where gamma in mathbbR, we conclude that any such affine combination yields f. Choosing gamma frac12, we obtain the symmetric part of mathrm A.

Positive definite matrices do not have to be symmetric it is just rather common to add this restriction for examples and worksheet questions. Though this restriction may seem a little severe, there are a number of important applications, which include some classes of partial differential equations and some classes of least squares problems. The advantage of this restriction is that the number of operations to do Gaussian elimination can be cut in half. You might find this previous question regarding non-symmetric positive definite matrices worthwhile reading: Does non-symmetric positive definite matrix have positive eigenvalues