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 ...
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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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Prove that the matrix $(I-xx^\top)(I-3 \mathrm{Diag}(xx^\top))(I-xx^\top)$ has at most one negative eigenvalue, for $x$ a unit vector
Let $x \in \mathbb{R}^n$ be a unit vector, i.e., $\|x\|=1$. Show that the following matrix $A$ has at most one negative eigenvalue:
$$A:=(I-xx^\top)(I-3 \mathrm{Diag}(xx^\top))(I-xx^\top).$$
$\mathrm{...
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Upper bound on the trace of the product of real PSD matrices involving the trace of one of them
I am looking at the negative log-likelihood of a multivariate Gaussian $X \sim \mathcal{N}(\mu, \Sigma)$:
$$const + (n/2)\log|\Sigma| + (1/2)Tr(\Sigma^{-1}S),$$
where $S = (1/n)\sum(X_i-\mu)(X_i-\mu)'$...
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Does subtracting a rank-1 matrix from a positive definite matrix result in positive semi-definite matrix? [closed]
Let $A$ be a symmetric positive definite matrix. We define the matrix $B$ as follows:
\begin{equation}
B = A - \frac{1}{e^\top A e} Aee^\top A
\end{equation}
where $e=(1,1,...,1)$.
I believe that $B$...
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Suppose block matrix $D$ in $\begin{pmatrix} A & B \\ C&D\\ \end{pmatrix}$ has Schur complement $S$. Is $S^{-1}-A^{-1}$ positive semi-definite? [closed]
Suppose that the positive semi-definite matrix $M$ is partitioned into four submatrices
blocks as
$$M = \begin{pmatrix} A & B \\ C&D\\ \end{pmatrix}$$
and block matrix $D$ has Schur complement
...
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How to calculate the eigenvalues of matrix with entries that are continuous functions of matrix?
I met the following problem during my research, could anyone please give me some help?
Let $A(x) \in \mathbb{R}^{2 \times 2}, x\in \mathbb{R}$ be a symmetric matrix in the form
\begin{align}
A(x) = \...
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Gram vectors after adding Identity
Suppose I have some matrix $G \in \mathbb{C}^{2 \times 4}$. Then the columns of $G$ are the Gram vectors of the positive semidefinite (psd) matrix $G^H G$ (Hermitian transpose of $G$ times itself). ...
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1
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Does the product of a PSD matrix and a positive eigenvalue matrix have positive eigenvalues?
It's known that if $A$ is a positive definite (symmetric) matrix and $B$ is positive semi-definite (not necessarily symmetric), then $AB$ has non-negative eigenvalues. It follows from knowing that $A$ ...
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Eigenvalues of a "diagonal" block-matrix
Let $n,n'\geq 1$ and $A_1 \in \mathbb{R}^{n \times n}, A_2 \in \mathbb{R}^{n' \times n'}$ be two symmetric matrices. Let $A = \begin{pmatrix}
A_1 & 0\\\
0 & A_2
\end{pmatrix} \in \mathbb{R}...
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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 ...
2
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2
answers
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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 ...
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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 $\Bbb R$. ...
0
votes
1
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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 ...
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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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For any $A\in \mathbb{R}^{n\times m} $, there exists a constant $a>0$ such that $A^TAA^TA-aA^TA \geq 0$.
Let $A\in \mathbb{R}^{n\times m}$ be an arbitrary matrix with any positive integer $n$ and $m$. I would like to show that there exists a positive constant $a>0$ such that
$$x^T(A^TAA^TA - aA^TA)x\...
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votes
3
answers
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Definiteness of symmetrix block matrix
Let $A$ and $D$ be symmetric positive definite matrices and consider the symmetric block matrix
$$ M := \begin{pmatrix} A & \alpha B \\ \alpha B^\top & D \end{pmatrix} $$
where $\alpha \in \...
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Equivalence between first-order optimality conditions for rank-$1$ constraint and quadratic form
When simplified down, my optimization problem has the following structure
$$\underset{x\in\mathbb{R}^n}{\arg\min} \quad \left\| b - A\operatorname{vec} \left( x x^\top \right) \right\|_2^2$$
I am ...
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votes
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answer
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Iterative algorithm for computing $\Sigma^{1/2} x$
Say I have a PSD matrix $\Sigma$ and a vector $x$, is there an iterative algorithm (faster than computing $\Sigma^{1/2}$ using Cholesky decomposition) for computing $\Sigma^{1/2} x$?
(In this ...
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Positive semidefinite block matrix
Let $M$ be an $n \times n$ block matrix defined as $$M = \begin{bmatrix} A & \mathbf{b} \\ \mathbf{b}^T & 1 \end{bmatrix}$$ where $A$ is an invertible and symmetric $(n-1) \times (n-1)$ matrix,...
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Norm of sum of positive matrices versus the sum of norms
Suppose that $P_1, \dots, P_k$ are positive $n \times n$ matrices.
Is it true that
$$ \sum_{k} \| { P_k } \| \leq n \left\| {\sum_k P_k} \right\|,$$
where $\| \cdot \|$ denotes the operator norm?
My ...
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1
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Change in kernel after rank-1 update
Let $B$ be an $n\times n$ symmetric and positive semidefinite matrix and $v\in \mathbb{R}^n$ a vector. Consider the positive semidefinite matrix $A=B+vv^T$. It is clear that $\mathrm{ker}(B)\cap (v)^\...
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Is the (kind of direct) sum of positive definite matrices (with common subdomain) positive definite?
Let $M_A : C \oplus A \,\tilde{\to} \, C \oplus A$ and $M_B : C \oplus B \,\tilde{\to} \, C \oplus B$ be two symmetric positive definite matrices whose domains share a common subspace $C$.
You can ...
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Hessian positive semidefinite for a matrix variable function?
I am currently reading a paper, and a proof (see below) in the paper talks about the hessian being positive semidefinite for a matrix variable function $g:\mathbb{R}^{n\times k} \to \mathbb{R}$, and ...
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If $\langle x,y\rangle>0$, does there exist a positive-definite transformation $A$ such that $Ax=y$?
In this answer, solving an exercise of Halmos, it is shown that:
If $x$ and $y$ are non-zero vectors and $\langle x,y\rangle$ is (real and) strictly positive, then there exists a positive semi-...
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Positive semidefiniteness in Boolean least-squares problem
A Boolean least-squares can be formulated as follows
\begin{array}{ll} \text{minimize} & \operatorname{tr}(A^TAX) - 2b^TAx + b^Tb\\ \text{subject to} & X = xx^T\\ & X_{ii} = 1\end{array}
...
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Why is gradient symmetric at optimal point for convex functions on the positive semidefinite cone?
For a convex function $f: \mathbb{R}^{n \times n} \to \mathbb{R}$, if $X^{*} = \underset{ X\succeq 0 }{ \operatorname{arg min}}f(X)$, then we have $$\nabla f(X^{*})\succeq 0, \qquad X^{*} \succeq 0, \...
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Span of positive (semi)definite matrices
In this answer, Ben Grossmann said that $\operatorname{span}\{P:P \succ 0\} = \{A:A = A^T\} =: \mathcal S$. I am not sure, however, why this is true. Also, I guess $\operatorname{span}\{ P:P\succeq 0 \...
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Matrix completion with Pfaffians constrain
Perform matrix completion of a real $2n \times 2n$ matrix $\mathbf{C}$ (i.e., we have only a few entries of the matrix $C$ and want to fill in the remaining ones) with the constraints that:
$\mathbf{...
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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 ...
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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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votes
2
answers
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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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votes
2
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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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1
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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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votes
1
answer
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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 ...
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1
vote
0
answers
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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 ...
3
votes
0
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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?...
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Are there any conditions for a product of PSD matrices to be PSD?
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 ...
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Preserving positive semi-definiteness
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 ...
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