# Questions tagged [positive-semidefinite]

Relating to a symmetric $n\times n$ real matrix $(M)$ such that the scalar $x^TMx\ge 0\ \forall x\in \Bbb{R}^n\backslash \{0\}$

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### Is the product of symmetric positive semidefinite matrices positive definite?

I see on Wikipedia that the product of two commuting symmetric positive definite matrices is also positive definite. Does the same result hold for the product of two positive semidefinite matrices? ...
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### Prove that every positive semidefinite matrix has nonnegative eigenvalues

There is a theorem which states that every positive semidefinite matrix only has eigenvalues $\ge0$ How can I prove this theorem?
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### Definiteness of a general partitioned matrix $\mathbf M=\left[\begin{matrix}\bf A & \bf B\\\bf B^\top & \bf D \\\end{matrix}\right]$

If $\mathbf M=\left[\begin{matrix}\bf A & \bf b\\\bf b^\top & \bf d \\\end{matrix}\right]$ such that $\bf A$ is positive definite, under what conditions is $\bf M$ positive definite, positive ...
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### Positive semidefinite cone is generated by all rank-$1$ matrices.

The positive semidefinite cone is generated by all rank-$1$ matrices $xx^T$, which form the extreme rays of the cone. Positive definite matrices lie in the interior of the cone. Positive semidefinite ...
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### The product of two symmetric, positive semidefinite matrices has non-negative eigenvalues

How can I prove the following? If $A$ and $B$ are two symmetric, positive semidefinite matrices then all eigenvalues of $AB$ are non-negative.
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### Checking if a matrix is positive semidefinite

Determine whether the following $2 \times 2$ matrix is positive semidefinite (PSD) $$\begin{bmatrix}\frac{2}{x} & \frac{-2y}{x^2} \\\frac{-2y}{x^2} & \frac{2y^2}{x^3}\end{bmatrix}$$ ...
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### Are positive definite matrices robust to “small changes”?

Let $A$ be a positive-definite matrix and let $B$ be some other symmetric matrix. Consider the matrix $$C=A+\varepsilon B.$$ for some $\varepsilon>0$. Is it true that for $\varepsilon$ small ...
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