# 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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### Negative definite forcing of an ODE ? (soft question)

Given an ODE in $\mathbb{R}^n$ $$\frac{d}{dt}x(t)=b(x(t)).$$ How can the assumption that $$\langle D b \xi,\xi\rangle \leq - c |\xi|,~~~~~\forall \xi \in \mathbb{R}^n$$ be interpreted? Does this stop ...
1 vote
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### Monotonicity of quadratic forms in positive semidefinite Matrices

I'm interested in some kind of monotonicity quality of quadratic forms. Let $A$ be a symmetric positive semidefinite (n x n)-matrix. Suppose we have two real n-dimensional vectors a and b, for which ...
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### Divergence theorem for the set of singular positive semidefinite matrices.

Let us consider the space of symmetric positive-definite (SPD) $\mathscr{P}^d$ matrices of dimensions $d \times d$. It is well known that this space is a pointed convex cone, which represents a ...
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### Distance between two positive semidefinite matrices

Given two positive semidefinite matrices X, and Y. My question is, can we use von Neumann or the log-Determinant divergences to measure the distance between the two matrices. Most Manifold measure ...
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### Difference of inverese matrices

Let $A$, $B$ be symmetric positive definite matrices. We know $$A^{-1}-B^{-1}=-C$$ for positive semidefinite matrix $C$. I want to show that $A-B$ is positive semidefinite. Here's my take: \begin{...
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### Proving that a symmetric real matrix with a specific structure is positive definite.

Let $H$ be an $N\times N$ symmetric matrix with the following structure: \begin{equation} [H]_{n,m}\triangleq\begin{cases} {\left|x_{n}\right|}, &\text{if}\ n=m,\\ -\mathsf{Re}{\left(...
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### Mercers condition and an RBF PSD-kernel.

I was given a question in an exam, and it made me realize i don't understand Mercer's condition quite well. I'd be happy for some insight about why my intuition is not right :) for the question we ...
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### When is $(A + A^\top)$ PSD even though A is not PSD?

I am wondering if there are any constraints which can be places on a matrix $A \in \mathbb{R}^{n \times n}$ to ensure that $A + A^\top$ is positive semidefinite (PSD) even though $A$ itself is not PSD....
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### Is there a Schur-like theorem for proving $B^TB\succeq A$?

This is in the context of semidefinite programs, and $A\in\mathbb R^{n\times n},~B\in\mathbb R^{k\times n}$ are variables. If want to show that $A-B^TB\succeq 0$ and I know $A\succeq 0$, then I can ...
1 vote
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### Is there a case in which PSD matrices $A$ and $B$ satisfy the condition $A \approx B$ but $A^2 \not \approx B^2$?

For two PSD matrices $A, B$ and a positive number $\alpha$, let's define $A \approx_{\alpha} B$ as $A \preceq \alpha B$ and $B \preceq \alpha A$, where $A \preceq B$ means $B - A$ is PSD. And if there ...
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### $A,B$ positive semidefinite matrices with $A \geq B$ implies $A^2 \geq B^2$? [duplicate]

Is it true that if I am considering positive semidefinite matrices $A, B$ with $A \geq B$ then $A^2 \geq B^2$? Could you help me prove this or think of a counterexample (eventually assuming the ...
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### Applying an affine transformation from the matrix to vector representations of spectahedra

On page 9, Parrilo wrote$^\color{magenta}{\star}$ that the vector representation of a spectrahedron $$S = \left\{ (x_1, \dots, x_m) \in {\Bbb R}^m : A_0 + \sum_{i=1}^m A_i x_i \succeq 0 \right\}$$ ...
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### On the transitivity of the Löwner order

If the Löwner order is a partial order, then it is transitive. If so, how can one prove it? Proposition. Let ${\Bbb S}_n ({\Bbb R})$ denote the set of $n \times n$ symmetric matrices over $\Bbb R$. ...
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### Question about proving positive definiteness

Good afternoon, I'm currently studying stochastic processes and would like to understand a step in a proof of the semi definiteness of the covariance function of the fractional Brownian Motion. A ...
1 vote
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### Is a non-negative quadratic polynomial always a sum of squares?

I am studying positive semi-definite matrices. A natural problem arises when we study the properties of the corresponding quadratic form of such matrices: Can we always represent such quadratic forms ...
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### Property of a matrix - $x^T A y\geq x^TAx$ implies $y^TAy>y^TAx$.

There is a $n\times n$ matrix $A$. $A$ satisfies the following property for $x,y\in \mathbb R^n$ and $x\neq y$, $$x^T A y\geq x^TAx\Rightarrow y^TAy>y^TAx.$$ Is there any known matrix property that ...
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### Is $f(\sigma^{-1/2} \rho \sigma^{-1/2}) = f(\sigma^{-1} \rho)$ true?

Let $\rho,\sigma$ are positive semidefinite operator, let $f: (0;+\infty) \rightarrow \mathbb{R}$ is operator convex function, in A. Lesniewski and M. B. Ruskai. Monotone Riemannian metrics and ...
1 vote
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### Largest and smallest eigenvalues of matrix with alternating 1s and 0s

My linear algebra is a bit rusty, so this may be a dumb question. But suppose I have an $N$-by-$N$ block matrix $A$, where each $(i,j)$ entry $A_{ij}$ is the $2\times 2$ identity matrix. Equivalently, ...
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### Perturbation of positive semidefinite matrix

Consider an $n\times n$ matrix $A$ that is positive semidefinite and has rank $n-1$, so there exists exactly one eigenvector $v$ such that $Av=0$. Let now $B$ be a symmetric matrix such that $v^TBv=0$....
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### Is $\left\{ A P + P A^{T} \mid P \succ 0 \right\}$ open?

Consider an $n$ by $n$ matrix $A$, and the set $$\left\{ A P + P A^{T} \mid P \succ 0 \right\}$$ Is this set open under the standard metric (the one induced by Frobenius norm) in the space of all ...
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### Are these Hankel matrices positive semidefinite?

While working on a quantum information project, I encountered the following two Hankel matrices $$a_{i,j} = (i+j)!(2n-(i+j))! ,\qquad b_{i,j} = (i+j+1)!(2n-(i+j))!$$ where $0 \le i,j \le n$ and $!$ ...
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### Positive definiteness of a product of three matrices [duplicate]

Let $A \in \mathbb{R}^{n \times m},$ with $1<m<n$, where $\text{rank}(A)=m$ (i.e., $A$ is a tall matrix of full column rank). Let $B \in \mathbb{R}^{n \times n}$ be a positive definite ...
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### Positive-definite block matrix with constant block sums

Given two natural numbers $n$ and $m$, suppose that $A$ is an $nm\times nm$ real nonnegative matrix. Seeing $A$ as a block matrix where each block has size $m\times m$, suppose that the sum of the ...
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### Will a positive semidefinite matrix $A$ become positive definite if a positive real value is added to one of its diagonal entries?

I have a symmetric positive semidefinite matrix $A$ that is irreducible, and with algebraic and geometric multiplicity for $\lambda(A) = 0$ as $1$. Another matrix $B$ with one entry in a diagonal is ...
1 vote
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### Simple representation of a positive linear transformation of a semidefinite cone

I am trying to solve a conic optimization problem where one of my length $n$ vector decision variables is the sum of all of the $n$ unique diagonal bands of any $n \times n$ semidefinite matrix. I can ...
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### Uniqueness of the square root for non-symmetric definite matrices

When $A$ is positive semi-definite, the positive semi-definite square root $B$, which is $BB=A$ is unique. I have heard that this holds true for a symmetric matrix. What about under weaker conditions?...
A matrix is positive semidefinite (PSD) if it's symmetric and all its eigenvalues are non-negative. There are many other equivalent definitions. Suppose $A$, $B$ are two PSD matrices. $AB$ is PSD if ...
Let $A, B\in \mathbb{R}^{n\times d}$ ($n \geq d$) be two matrices with $A^\top A \preceq B^\top B$. Let $D = \mathrm{diag}(d_1, \ldots, d_n)$ be a diagonal matrix with nonnegative entries. Do we still ...