Question

Is the property of positive definiteness of a matrix restricted only to a symmetric matrix ?

Answer 1

Confusingly, the discussion of positive definite matrices is often restricted to only Hermitian matrices, or symmetric matrices in the case of real matrices (Pease 1965, Johnson 1970, Marcus and Minc 1988, p. 182; Marcus and Minc 1992, p. 69; Golub and Van Loan 1996, p. 140). A Hermitian (or symmetric) matrix is positive definite iff all its eigenvalues are positive. Therefore, a general complex (respectively, real) matrix is positive definite iff its Hermitian (or symmetric) part has all positive eigenvalues.

reference: http://mathworld.wolfram.com

Answer 2

A matrix can be positive definite but not symmetric. For example:
\[A=\left[\begin{matrix}1 & 1 \\ -1 & 1\end{matrix}\right]\]
is positive definite but it is not symmetric.