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== Extension for non-Hermitian square matrices ==
The definition of positive definite can be generalized by designating any complex matrix <math>M</math> (e.g. real non-symmetric) as positive definite if <math>\Re\left(\mathbf{z}^* M\mathbf{z}\right) > 0</math> for all non-zero complex vectors <math>\mathbf z</math>, where <math>\Re(c)</math> denotes the real part of a [[complex number]] <math>c</math>.<ref name="mathw">Weisstein, Eric W. ''[http://mathworld.wolfram.com/PositiveDefiniteMatrix.html Positive Definite Matrix.]'' From ''MathWorld--A Wolfram Web Resource''. Accessed on 2012-07-26</ref> Only the Hermitian part <math display="inline">\frac{1}{2}\left(M + M^*\right)</math> determines whether the matrix is positive definite, and is assessed in the narrower sense above. Similarly, if <math>\mathbf x</math> and <math>M</math> are real, we have <math>\mathbf{x}^\operatorname{T} M \mathbf{x} > 0</math> for all real nonzero vectors <math>\mathbf x</math> if and only if the symmetric part <math display="inline">\frac{1}{2}\left(M + M^\operatorname{T}\right)</math> is positive definite in the narrower sense. It is immediately clear that <math display="inline">\mathbf{x}^\operatorname{T} M \mathbf{x} = \sum_{ij} x_i M_{ij} x_j</math>is insensitive to transposition of ''M''.
Consequently, a non-symmetric real matrix with only positive eigenvalues does not need to be positive definite. For example, the matrix <math>M = \left[\begin{smallmatrix} 4 & 9 \\ 1 & 4 \end{smallmatrix}\right]</math> has positive eigenvalues yet is not positive definite; in particular a negative value of <math>\mathbf{x}^\operatorname{T} M\mathbf{x}</math> is obtained with the choice <math>\mathbf{x} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right] </math> (which is the eigenvector associated with the negative eigenvalue of the symmetric part of {{nowrap|<math>M</math>).}}
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