positive semi-definite matrix and determinant

**altave86** · September 27th 2009, 03:31 AM

Having a positive determinant is one of the main properties of a Postive Semi-Definite matrix. How do I prove it?

**NonCommAlg** · September 27th 2009, 09:58 PM

Originally Posted by altave86

Having a positive determinant is one of the main properties of a Postive Semi-Definite matrix. How do I prove it?

let $\lambda$ be an eigenvalue of $A.$ then $A\bold{x}=\lambda \bold{x},$ for some $\bold{x} \neq \bold{0}.$ thus $0 \leq \bold{x}^* A \bold{x}=\lambda \bold{x}^* \bold{x}=\lambda ||\bold{x}||^2$ and so $\lambda \geq 0.$ the result now follows because $\det A$ is the product of the eigenvalues of $A.$

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