In some old course notes I'm reading to touch up on statistical forecasting methods, the book often makes reference to "non-negative definite" matrices. I know what a semi-positive definite, positive definite, and indefinite matrix are, but I've never heard this terminology before. Further, online resources don't really seem to mention it, so I'm unsure what exactly it is equivalent to as the "non-negative definite" property isn't directly applied anywhere. Any ideas?
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$\begingroup$Non-negative definite is a synonym for positive semidefinite. One thing you should watch out for, however, is that while most authors restrict these terms to real symmetric or hermitian matrices, not all do so.
$\endgroup$ 2 $\begingroup$An $n\times n$ matrix $A$ is non-negative definite (aka positive semi-definite) provided $x^tAx\ge 0$ for each column vector $x$ of length $n$.
$\endgroup$ $\begingroup$An $n\times n$ real symmetric matrix A is non-negative definite (aka positive semi-definite) provided $x^TAx\geq 0$ for all $x\in\mathbb{R}^n$ where $x^T$ is the transpose of $x$.
An $n\times n$ complex symmetric matrix A is non-negative definite (aka positive semi-definite) provided $x^*Ax\geq 0$ for all $x\in\mathbb{C}^n$ where $x^*$ is the complex conjugate of $x$.
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