An n × n real symmetric matrix M is positive definite if z^{T}Mz > 0 for all nonzero vectors z with real entries (i.e. z ∈ R^{n}), where z^{T} denotes the transpose of z.
For complex matrices, this definition becomes: a Hermitian matrix is positive definite if z^{*}Mz > 0 for all nonzero complex vectors z, where z^{*} denotes the conjugate transpose of z. The quantity z^{*}Mz is always real because $M$ is a Hermitian matrix. For this reason, positivedefinite matrices are often defined to be Hermitian matrices satisfying z^{*}Mz > 0. The section NonHermitian matrices discusses the consequences of dropping the requirement that M be Hermitian.
1.  All eigenvalues $lambda\_i$ of $M$ are positive. Recall that any Hermitian M, by the spectral theorem, may be regarded as a real diagonal matrix D that has been reexpressed in some new coordinate system (i.e., $M\; =\; P^\{1\}DP$ for some unitary matrix P whose rows are orthonormal eigenvectors of M, forming a basis). So this characterization means that M is positive definite if and only if the diagonal elements of D (the eigenvalues) are all positive. In other words, in the basis consisting of the eigenvectors of M, the action of M is componentwise multiplication with a (fixed) element in C^{n} with positive entries. 
2.  The sesquilinear form

3.  M is the Gram matrix of some collection of linearly independent vectors
for some k. That is, M satisfies:
The vectors x_{i} may optionally be restricted to fall in C^{n}. In other words, M is of the form A*A where A is not necessarily square but must be injective in general. 
4.  All the following matrices have a positive determinant (the Sylvester criterion):
In other words, all of the leading principal minors are positive. For positive semidefinite matrices, all principal minors have to be nonnegative. The leading principal minors alone do not imply positive semidefiniteness, as can be seen from the example

5.  There exists a unique lower triangular matrix $L$, with strictly positive diagonal elements, that allows the factorization of $M$ into

For real symmetric matrices, these properties can be simplified by replacing $mathbb\{C\}^n$ with $mathbb\{R\}^n$, and "conjugate transpose" with "transpose."
is a bilinear map such that B(x, y) is always the complex conjugate of B(y, x). Such a function B is called positive definite if B(x, x) > 0 for every nonzero x in V.
for all nonzero $x\; in\; mathbb\{R\}^n$ (or, equivalently, all nonzero $x\; in\; mathbb\{C\}^n$).
It is called positivesemidefinite if
for all $x\; in\; mathbb\{R\}^n$ (or $mathbb\{C\}^n$).
It is called negativesemidefinite if
for all $x\; in\; mathbb\{R\}^n$ (or $mathbb\{C\}^n$).
A matrix M is positivesemidefinite if and only if it arises as the Gram matrix of some set of vectors. In contrast to the positivedefinite case, these vectors need not be linearly independent.
For any matrix $A$, the matrix A^{*}A is positive semidefinite, and rank($A$) = rank(A^{*}A). Conversely, any positive semidefinite matrix $M$ can be written as M = A^{*}A; this is the Cholesky decomposition.
A Hermitian matrix which is neither positive nor negativesemidefinite is called indefinite.
A matrix is negative definite if all kth order leading principal minors are negative if k is odd and positive if k is even.
For arbitrary square matrices $M,N$ we write $Mgeq\; N$ if $MN\; geq\; 0$, i.e. $MN$ is positive semidefinite. This defines a partial ordering on the set of all square matrices. One can similarly define a strict partial ordering $M>N$.
1.  Every positive definite matrix is invertible and its inverse is also positive definite. If $M\; geq\; N\; >\; 0$ then $N^\{1\}\; geq\; M^\{1\}\; >\; 0.$ 
2.  If $M$ is positive definite and $r\; >\; 0$ is a real number, then $r\; M$ is positive definite. If $M$ and $N$ are positive definite, then the sum $M\; +\; N$ and the products $MNM$ and $NMN$ are also positive definite. If $M\; N\; =\; N\; M$, then $M\; N$ is also positive definite. 
3.  M=(m_{ij}) > 0 then the diagonal entries $m\_\{ii\}$ are real and positive. As a consequence $text\{tr\}(M)>0$. Furthermore

4.  A matrix $M$ is positive definite, if and only if there is a positive definite matrix $B>0$, called the square root of B, with $B^2\; =\; M$. One writes $B\; =\; M^\{1/2\}$. This matrix $B$ is unique (but only under the assumption $B>0$). If $M\; >\; N\; >\; 0$ then $M^\{1/2\}\; >\; N^\{1/2\}>0$. 
5.  If $M,N\; >\; 0$ then $Motimes\; N\; >\; 0.$ (Here $otimes$ denotes Kronecker product.) 
6.  For matrices $M=(m\_\{ij\}),N=(n\_\{ij\})$ write $Mcirc\; N$ for the entrywise product of $M$ and $N$, i.e. the matrix whose $i,j$ entry is $m\_\{ij\}\; n\_\{ij\}$. Then $M\; circ\; N$ is the Hadamard product of $M$ and $N$. If $M,N>0$ then $Mcirc\; N\; >\; 0$ and if $M,N$ are real matrices, the following inequality, due to Oppenheim, holds: $det(Mcirc\; N)\; geq\; (det\; N)\; prod\_\{i\}\; m\_\{ii\}.$ 
7.  Let $M\; >\; 0$ and $N$ Hermitian. If $MN+NM\; geq\; 0$ ($MN+NM\; >\; 0$) then $Ngeq\; 0$ ($N\; >\; 0.$ ) 
8.  If $M,Ngeq\; 0$ are real matrices then $text\{tr\}(MN)geq\; 0.$ 
9.  If $M>0$ is real, then there is a $delta>0$ such that $Mgeq\; delta\; I$ where $I$ is the identity matrix. 
A real matrix M may have the property that x^{T}Mx > 0 for all nonzero real vectors x without being symmetric. The matrix
satisfies this property, because for all real vectors $x\; =\; (x\_1,\; x\_2)^T$ such that $x\; ne\; 0$,
In general, we have x^{T}Mx > 0 for all real nonzero vectors x if and only if the symmetric part, (M + M^{T}) / 2, is positive definite.
The situation for complex matrices may be different, depending on how one generalizes the inequality z^{*}Mz > 0. If z^{*}Mz is real for all complex vectors z, then the matrix M is necessarily Hermitian. So, if we require that z^{*}Mz be real and positive, then M is automatically Hermitian. On the other hand, we have that Re(z^{*}Mz) > 0 for all complex nonzero vectors z if and only if the Hermitian part, (M + M^{*}) / 2, is positive definite.
In summary, the distinguishing feature between the real and complex case is that, a bounded positive operator on a complex Hilbert space is necessarily Hermitian, or self adjoint. The general claim can be argued using the polarization identity. That is no longer true in the real case.
There is no agreement in the literature on the proper definition of positivedefinite for nonHermitian matrices.