# 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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### 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 ...
3answers
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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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### Meaning of $x^T A x$

I've seen the term $x^T A x$ come up in a bunch of different areas of linear algebra, where A is a square and usually symmetric matrix. Places I've seen it include defining the Raleigh quotient, ...
3answers
338 views

### Determining if a symmetric matrix is positive definite

I have a symmetric matrix where all non-diagonal elements are positive and identical, and all diagonal elements are identical as well. For example, the $3 \times 3$ version of this matrix has the ...
1answer
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### Matrices whose all principal $k\times k$ sub-matrices are positive semidefinite

I would like to know whether the set of $n\times n$ Hermitian matrices whose all ${{n}\choose{k}}$ principal $k\times k$ sub-matrices---the matrices obtained by removing $n-k$ columns as well as the ...
1answer
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### When is $\det(A+B)=\det(A)+\det(B)$ for positive definite $A$ and positive semidefinite $B$?

Let $A$ be a positive definite matrix and $B$ be a positive semi-definite matrix. Under what conditions does $\det(A+B)=\det(A)+\det(B)$ hold? We all know that for any two positive semi-definite ...
1answer
259 views

### Variables that will make this matrix positive semi-definite

I have a matrix $$M=\begin{bmatrix} 1+t+m &n&t+n&m+c \\ n &1+t-m&m-c & t-n \\ t+n & m-c&1-t-m & -n \\ m+c & t-n & -n & 1-t+m \end{bmatrix}$$ where I ...
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1answer
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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 ...
2answers
779 views

### When does a real, positive definite matrix have positive entries?

Let $A = (a_{ij})_{i,j=1,\dots,n}$ be a real symmetric square matrix. Suppose $A$ is positive definite. Are there sufficient conditions that guarantee $a_{ij} > 0$ for all $i,j = 1,\dots, n$? I ...
1answer
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### Matrices Inequality Proof

Recently, I read a paper and there is a step which turns out not obvious to me. The statement is as follows: All matrices here are real matrices. $F$ is an arbitrary square matrix. $\Psi$ is a ...
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### Show that the matrix $\big(\frac{1}{a_i + a_j}\big)_{n\times n}$ is positive semidefinite [duplicate]

Given positive real numbers $a_1,\dots ,a_n$, how can I prove that the symmetric matrix composed of the entries $\frac{1}{a_i + a_j}$ is positive semi-definite *this is not a hw question but a self ...
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### Checking positive semidefiniteness in MATLAB

Let $\mathbf{A}$ be a $n\times n$ matrix. I want to check in MATLAB if it is PSD or not. Which tests, in MATLAB, should I do for this purpose? I know that if $\mathbf{A}$ is PSD then following holds ...
5answers
375 views

### Show that if $A$ is positive definite then $A + A^{-1} - 2I$ is positive semidefinite

Let $A$ be a real symmetric positive definite matrix. Show that $$A + A^{-1} -2I$$ is positive semidefinite. I found that $A^{-1}$ is a positive definite matrix, thus $A + A^{-1}$ is also a positive ...
2answers
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### Ensuring that a symmetric matrix with nonnegative elements is positive semidefinite

I have the following matrix $A$: symmetric all positive and/or zero values the main diagonal is all the same value, $x$. To ensure that the matrix $A$, is positive semidefinite, must I only ensure ...