# Questions tagged [positive-semidefinite]

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

429 questions
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### Intuitive explanation of a positive semidefinite matrix

What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is ...
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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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### 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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### 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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### Does a positive semidefinite matrix always have a non-negative trace?

A simple question: If $A$ is a positive semidefinite matrix ($A\succeq 0)$, does it imply that $\mbox{Tr}(A)\geq 0$, where the $\mbox{Tr}(\cdot)$ denotes the trace. If not, any counter-example? ...
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### Probability that a random binary matrix is positive semi-definite

Let $A$ be a random $n \times n$ matrix such that $A_{ij}\in\{0,1\}$. Assume that each element $A_{ij}$ equals 1 with some probability $p>0$ and that all the draws are independent across elements. ...
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### Difficulty in linear algebra question.

Let $a_{ij}=a_ia_j$ $1\leq i,j\leq n$ where $a_1,... a_n$ are real numbers. Let $A=(a_{ij})$ be the $n \times n$ matrix . Then It is spoosible to choose $a_1,... a_n$ so as to mke A non singular. ...
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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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### 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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### Criterion for positive semidefinite matrices

Is there a criterion for positive semidefiniteness of a matrix in terms of dimension reduction, i.e, such that positive semi-definiteness of $n \times n$ matrix is expressed as positive ...
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### Is $U=V$ in the SVD of a symmetric positive semidefinite matrix?

Consider the SVD of matrix $A$: $$A = U \Sigma V^\top$$ If $A$ is a symmetric, positive semidefinite real matrix, is there a guarantee that $U = V$? Second question (out of curiosity): what is the ...
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### Counter-example for a matrix not being a correlation matrix

Is there an example of an $n \times n$ matrix that: is real-valued and symmetric with entries between $-1$ and $+1$ has diagonal elements equal to $1$ has a non-negative determinant has non-negative ...
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### Is the function $\mbox{tr}(XAX')$ convex?

Is the function $\mbox{tr}(XAX')$ convex, where $A$ is a positive semidefinite (PSD) matrix? I know that for a general $A$, the above trace function is not convex. But for a PSD $A$, is the function ...
I know that any local minimum $x_0$ of a (twice continuously differentiable) function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ has positive semi-definite Hessian $H(x_0) \succeq 0$. Can we say ...