TITLE: Tests for positive definiteness of nonsymmetric matrices
QUESTION [6 upvotes]: Do tests for positive definiteness for nonsymmetric matrices exist?  More specifically I am working with bidiagonal upper/lower triangular matrices with positive eigenvalues and I need to check to see if they are positive definite or not.

REPLY [10 votes]: Suppose that $A$ is real.  Note that $x^TAx=(x^TAx)^T=x^TA^Tx$, being a scalar.  Hence $x^TAx>0$ if and only if $2x^TAx=x^TAx+x^TA^Tx=x^T(A+A^T)x>0$.  Thus $A$ is positive definite if and only if $A+A^T$, a symmetric matrix, is positive definite.  You can test this using the Cholesky decomposition.
If $A$ is complex, replace $x^TAx$ with $x^\star Ax$ and $A^T$ with $A^\star$.