I'm doing a work on covariance matrix and this question came to me that was not very clear.
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1Related questions: Is every covariance matrix positive definite? considers the broader case of covariance matrices, of which correlation matrices are a special case; also Is every correlation matrix positive semi-definite? and Is every correlation matrix positive definite? – Sycorax Jun 23 '22 at 02:07
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Also https://stats.stackexchange.com/questions/69114/why-does-correlation-matrix-need-to-be-positive-semi-definite-and-what-does-it-m – Sycorax Jun 23 '22 at 02:08
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An $n\times n$ matrix $A$ is said to be positive definite if $\mathbf{x}^T A \mathbf{x}>0 \>\forall \> \mathbf{x} \neq \mathbf{0} \in \mathbb{R}^n$. The term "positive definite" just refers to a subclass of matrices.
Demetri Pananos
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2If you replace $\gt$ by $\ge$ then you have the definition of positive semidefinite – Henry Jun 23 '22 at 07:43