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For questions about positive definite real or complex matrices. For questions about positive semi-definite matrices, use the (positive-semidefinite) tag.

This tag is for questions about positive definite matrices with real or complex entries. A square matrix $M \in \mathbf{F}^{n \times n}$ ($\mathbf{F} = \mathbf{R}$ or $\mathbf{C}$) is positive definite if $$ \text{for all } x \in \mathbf{F}^n \setminus \{0\}, x^\dagger M x > 0. \tag{1}$$ Here $x^\dagger$ denotes the transpose if $x$ is real and the conjugate-transpose if $x$ is complex.

If we replace $(1)$ with $$ \text{for all } x \in \mathbf{F}^n, x^\dagger M x \ge 0 $$ then $M$ is said to be positive semi-definite. All positive definite matrices are positive semi-definite. Questions about positive semi-definite matrices not specifically about positive definite matrices should use the tag instead or in conjunction.

If $\mathbf{F} = \mathbf{C}$ then $M$ is positive definite if and only if $M^\dagger = M$ and every eigenvalue of $M$ is a positive real number. If $\mathbf{F} = \mathbf{R}$ then it is not necessary that $M^\dagger = M$, for instance $$ M = \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix} $$ is a positive definite real matrix but not symmetric. Some authors require that a positive definite matrix be symmetric.

Some authors use a weaker form of $(1)$, namely $$ \text{for all } x \in \mathbf{F}^n \setminus \{0\}, \operatorname{Re}(x^\dagger M x) > 0. $$ With this definition it is no longer necessary that $M^\dagger = M$, even if $\mathbf{F} = \mathbf{C}$.