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Questions tagged [positive-semidefinite]

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

If $M$ is a positive semidefinite matrix then it has some additional properties which can be found in this Wikipedia article. You can also use this tag if one or more of these properties leads back to $M$ being positive-semidefinite.

1203 questions
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Proving matrix is semi-definite if diagonal entries are maximal

Suppose I have a symmetric real matrix with only entries that are $\ge 0$ and I know the entries in the diagonal are larger or equal to any other entry in the row (and because of the symmetry, this also goes for the column). Note that it is not a…
yagod
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Matrix with no negative elements = Positive Semi Definite?

A matrix $A$ is positive semi-definite IFF $x^TAx\geq 0$ for all non-zero $x\in\mathbb{R}^d$. If all elements of $A$ are non-negative, does this guarantee that $A$ is positive semi-definite?
24n8
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Why is positive semi-definiteness for matrices analgous to positiveness for numbers?

I am reading a textbook which states for $A, B$ symmetric, $A \leq B$ iff $B-A$ is non-negative definite (equiv. positive semi-definite). This is easily seen to be an ordering. This reads like the following statement for real numbers, applied to…
mai
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positive semidefinite matrix for 3x3 case

Given a symmetric matrix $$ A = \begin{bmatrix} x_1 & x_2 & x_3 \\ x_2 & x_4 & x_5 \\ x_3 & x_5 & x_6\end{bmatrix} $$ we want to find the conditions on these 6 variables that make the matrix positive semidefinite. Now, a part of the explanation…
manav
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If $B^{m\times m}$ is positive semidefinite, is $A^TBA$ is positive semidefinite?

If $B^{m\times m}$ is positive semidefinite, is it true that $A^TBA$ is positive semidefinite with $A^{m\times n}$? I think it is, because all of my counterexamples have failed, but I don't know how to prove it. I know that $x^TBx\ge 0$ if B is…
Vons
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Is difference between two positive semi definite (p.s.d) matrices still p.s.d?

I am working on two p.s.d matrices of the form $A=aa^T$ and $B=bb^T$ where $b^Tb>a^Ta$. So both $A$ and $B$ are p.s.d matrices of rank 1. Can I claim $(B-A)$ is also p.s.d? Or under what conditions that $(B-A)$ is p.s.d? Meanwhile, For any two…
Charles Chou
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Proof of why $\mathbf{A}_{ij}=i(n-j+1)$ for $j \geq i$ indicates positive semidefiniteness

Problem When reading an advanced text in numerical computing, I encountered the following claim If a symmetric matrix $\mathbf{A}\in \mathbb{R}^{n\times n}$ has entries $\mathbf{A}_{ij}=i(n-j+1)$ for $j \geq i$, then it is positive semidefinite. I…
Mr.Robot
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Semidefinite Matrix in LINGO

Using LINGO, I need to enter the following block matrix as one of my constraints $ M= \left[ {\begin{array}{cc} 1 & x^T \\ x & X \\ \end{array} } \right] $ where x is an n by 1 column matrix and X is an n by n symmetric matrix. Then I…
user321821
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How to prove a matrix is positive semidefinite?

Let $X\in S^3_+$ be a semidefinite cone. Show the explicit conditions on the components of $X$. I wanted to show for a positive semidefenite matrix $X$ we have $z^T Xz\geq0\forall z$: $$\begin{bmatrix} z_1& z_2& z_3 \end{bmatrix}\begin{bmatrix} x_1&…
havakok
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How can I derive the conditions of Positive semidefinite cone in $2\times2$ matrix.

By the definition, in order for $X$ to be positive semidefinite cone in $S^2$, it should satisfy that \begin{equation} X=\left[ \begin{array}{cc} x & y \\ y & z \end{array} \right]\in S_+^2 \quad\Longleftrightarrow\quad x\ge0,\quad z\ge0, \quad…
Danny_Kim
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Eigenvalues and positive semidefiniteness of a special matrix

Consider the following $(n+1)\times(n+1)$ real matrix $$A=\begin{pmatrix}a&p^t\\p&D\end{pmatrix},$$ where $D$ is an $n\times n$ diagonal matrix with strictly positive entries, $a>0$, and also elements of the vector $p$ are non-negative (some could…
Poor.Jack
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Is $xx^t+yy^t-uu^t$ diagonal?

Let $x, y\in \mathbb{R}^n$ are two given vectors. Can I always find a $u\in\mathbb{R}^n$ such that $$xx^t+yy^t-uu^t$$ is a diagonal matrix with non-negative elements? I could prove it when $n=2$, but don't know for higher $n$. Probably it would not…
Poor.Jack
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Singular matrix and positive semidefiniteness

Let $C_x$ be a psd matrix. If $b^TC_xb=0$, then why must $C_x$ be a singular matrix and how do you prove it?
eet
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Hessian Matrix is positive semidefinite

Let $\alpha$ and $\beta$ be vectors in $\mathbb{R}^n$, and $\beta^tx>0$ for all $x$ in $\mathbb{R}^n$, the Hessian Matrix is $\{H(x)\}_{ij}=\frac{2}{(\beta^t x)^3}\left((\alpha^t x)^2(\beta_i\beta_j)+(\beta^t x)^2(\alpha_i\alpha_j)-(\alpha^t…
user786004
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Is negative square root defined for a positive semidefinte matrix?

If I have a positive semidefinite matrix $A$, is an operation like $A^{-\frac{1}{2}}$ defined? I know it is defined for positive definite matrices but for the positive semidefinite case, do the zero eigenvalues cause trouble or do we just consider…
RandomMatrices
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