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Questions tagged [positive-semidefinite]

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

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29 views

Can i prove that this matrix is PSD?

I have matrix $A \in \mathbb{R}^{N \times N}$, such that $A(i,j)=trace(B_iCB_j), \forall ij$. Matrices $B_i$ and C are PSD and symmetric with positive entries. Can I prove that $A$ is PSD too? In ...
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0answers
17 views

Why can a constraint on a matrix being positive definite be rewritten as the matrix minus the identity being positive semidefinite?

My instructor today mentioned that if we have a constraint that a matrix $A$ is positive definite, then we can rewrite this constraint as $A - I$ is positive semidefinite without this affecting the ...
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1answer
37 views

Why taking integral of both sides of matrix inequality is allowed?

How to show if $\nabla^2 f(x) \succeq \alpha I$, then the function is $\alpha$-strongly convex? In my optimization notes I have $$\nabla^2 f(x) \succeq \alpha I \rightarrow \alpha\text{-strongly ...
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1answer
31 views

How can i prove the following function is positive?

I have the following function. $F =[x_1,x_2,...,x_n]_{1 \times n}*M_{n \times n}*([\dfrac{x_1}{|x_1|^{1/2}},\dfrac{x_2}{|x_2|^{1/2}}, ..., \dfrac{x_n}{|x_n|^{1/2}}]^T)_{n \times 1}$ Which $x \in \...
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1answer
30 views

How to show $\text{Tr}(AB) \leq \text{Tr}(AC)$ where $B \preceq C$?

Given three positive semi-definite matrices $A, B, C$. Show $\operatorname{Tr}(AB) \leq \operatorname{Tr}(AC)$ where $B \preceq C$? This inequality is the matrix form of multiplying a positive ...
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1answer
29 views

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{...
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1answer
21 views

Is this matrix positive semidefinite (Symmetric matrix, with particular pattern)

Let's consider a symmetric matrix A. If for each row, the diagonal entry is equal or larger than the magnitude of any other element, that is $$a_{ii} \geq |a_{ij}| \quad\text{for all rows } i \text{...
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1answer
57 views

What is the formula for projection onto spectraplex?

A spectraplex (special case of spectrahedron) is the set of all positive semi-definite matrices whose trace is equal to one. Formally, let $$ S=\{\textbf{W} \in \mathbb{R}^{d \times d} \mid \textbf{W} ...
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2answers
39 views

The proof of positive semi-definite for a kernel

How to prove the following kernel $K$ over $\mathbb R \times \mathbb R$ is positive semi-definite: $$K(x_i, x_j) = e^{-\lambda[\sin(x_i - x_j)]^2},$$ where $\lambda > 0$. It looks like the gaussian ...
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1answer
21 views

If matrices $A$ and $AB$ have full column rank, how do I prove that $P_A - P_{AB}$ is positive semidefinite?

First of all, the projection matrix $P_A$ is given by $P_A = A(A'A)^{-1}A'$. Similarly, $P_{AB} = AB(B'A'AB)^{-1}B'A'$. I have tried proving that $P_A - P_{AB}$ is itself a projection matrix, then it ...
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13 views

Positive definite squares [duplicate]

Suppose that $A, B$ are real $n\times n$ symmetric positive definite matrices such that $A - B$ is positive semi-definite. Does it follow that $A^2 - B^2$ is positive semi-definite?
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75 views

1-norm and symmetry

Define the fidelity function for positive operators by $F(\rho, \sigma) = \lVert \sqrt{\rho}\sqrt{\sigma}\rVert_1$. Here, $\lVert\cdot\rVert_1$ is the Schatten 1-norm and defined as $\lVert A\rVert_1 =...
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62 views

Convexity of a Log Likelihood Function

Goal I would like to proof than the Negative Log Likelihood Function of Sample drawn from a Normal Distribution is convex. Below a Figure showing an example of such function: Motivation of this ...
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7 views

Nonnegative Fourier series coefficients for periodic nonnegative-definite function

Is there a simple way to show that the Fourier series coefficients of a periodic, nonnegative-definite function $\kappa$ must all be nonnegative? (By nonnegative-definite I mean that the Gram matrix $\...
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18 views

Intervals of a Multivariable Function

If the gradient at some point of a multivariable function equals $\vec{0}$, and the Hessian is positive or negative semidefinite, is there a notion, as in single variable calculus, of resolving the ...
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1answer
28 views

Positive/Negative Definite/Semidefinite Test Generality

A test to determine whether a matrix is positive definite, negative definite, positive semidefinite, negative semidefinite, or none of the above, is to calculate the determinant of every cascading ...
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0answers
59 views

Is the matrix $a_{ij} = \frac{1}{i+j-1}$ Positive Definite? [duplicate]

I have a matrix $A$ of size $n \times n$ defined as $$a_{ij} = \frac{1}{i+j-1}$$ Where $a_{ij}$ is the $i^{th}$ row, $j^{th}$ column element of the matrix. So, $1 \leq i \leq n$ and $1 \leq j \leq n$....
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1answer
23 views

Ellipsoid representation by PSD matrix and by linear mapping

Consider the following two representations of ellipsoid: $$E_1 = \{x \mid x^TSx\leq 1, \, S \succ 0\}$$ and $$E_2 = \{y \mid y = Ax, \, \|x\|\leq 1, \, \det(A) \neq 0\}$$ If I want $E_1=E_2$, I ...
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1answer
12 views

Prove $y^tH_f(a)y \leq 0$ with Taylors Theorem

Let the function $f \in C^2(\mathbb{R}^n;\mathbb{R})$ have a local maximum in the point $a \in \mathbb{R^n}$. How can one prove the following with Taylor's theorem: The following applies: $y^tH_f(a)...
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1answer
18 views

Find closet PSD matrix to given diagonal matrix

Let's say I have an arbitrary diagonal matrix $A$. I want to convert this diagonal matrix to the closest matrix by some metric that is also PSD. Is there a standard way to do this? This is for a ...
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0answers
12 views

Sum of degenerate quadratic forms.

I am searching for an analogue of the fact: let $\Sigma_1 , \Sigma_2> 0$ in $\mathbb R^{m \times m}$ and let $x,c_1, c_2 \in \mathbb R^m$ be arbitrary. Let $\Sigma_3^{-1} = \Sigma_1^{-1} + \Sigma_2^...
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0answers
29 views

Inverse of Graph Laplacian does not exist in Matlab

In MATLAB, I have the following Laplacian matrix: $$L = \begin{bmatrix} 11&-5&-6&0&0&0\\ -5&20&-8&-3&-4&0\\ -6&-8&20&0&-6&0\\ 0&-3&...
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41 views

Condition for a symmetric matrix to contain only positive entries?

I have a matrix defined as $\mathbf{X} -\mathbf{Y}\mathbf{Y}^\top$, $\mathbf{Y}$ is a matrix containing only non-negative entries and $\mathbf{X}$ is a similarity matrix, hence $\mathbf{X}$ also ...
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1answer
50 views

If a matrix $Q$ is symmetric and positie definite, is it possible to show that the matrix $Q-A^T(AQ^{-1}A^T)^{-1}A$ is also positive definite?

If I have a symmetric and positive definite $n\times n$ matrix $Q$ and a full row-rank totally unimodular $m\times n$, where $m<n$, matrix $A$, is it posible to show that the matrix $$Q-A^T(AQ^{-...
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24 views

The Riccati equation and its asymptotic behavior

Consider matrices $A\in\mathbb{R}^{n\times n},B\in\mathbb{R}^{n\times m}$, a positive semidefinite symmetric matrix $Q\in\mathbb{R}^{n\times n}$ and a positive definite symmetric matrix $R\in\mathbb{R}...
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2answers
47 views

Positive semidefiniteness of symmetric matrix with diagonal = 1 and non-diagonal elements less than 1

Does anyone know any useful results with respect to symmetric matrices with constant diagonals (specifically with respect to whether all eigenvalues are greater than $0$)? I am working on a set of ...
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0answers
15 views

Positive semidefinite block matrix is PSD with negated off-diagonal blocks?

I am trying to follow some notes that imply $$ M= \begin{bmatrix} A&B^T\\B&C \end{bmatrix} \succeq 0 \Longleftrightarrow M'= \begin{bmatrix} A&-B^T\\-B&C \end{bmatrix} \succeq 0$$ and ...
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1answer
49 views

Decomposing a positive semi-definite matrix with all -1,+1 elements

Claim.$\,$ A matrix $\,X \in \{-1,1\}^{n\times n}\,$ is positive semi-definite if and only if it is of the form $X= xx^{T}$, for some $x \in \{-1,1 \}^n$. How can I prove this? Proving the 'if' part ...
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1answer
45 views

Let $A$ be a positive definite matrix and $B$ be a positive semi-definite matrix. Then $AB$ is diagonalizable. [closed]

Let $A$ be a positive definite matrix and $B$ be a positive semi-definite matrix. Then $AB$ is diagonalizable. I want to see if this is true or false.
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0answers
20 views

Help with matrix selection in matlab

I have this equalities, with $\Delta X$ and $\Delta S$ unknown matrices, $X$, $S$ and $\tau$ known. $$W^{-1}\Delta X W^{-1}+\Delta S=\tau X^{-1}-S$$or equivalently $$\Delta X + W\Delta S W=\tau S^{-1}...
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Prove the following set of unit vectors is orthogonal.

Suppose ${v_1 , v_2 , ……..v_n}$ are unit vectors in $\mathbb R^n$ such that $ || v||^2 = \sum_{n=1}^{\infty} | <v_i , v>|^2 $ for all $v \in\mathbb R^n $ Then I have to prove that ...
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1answer
26 views

Help with Matlab problem with positive semidefinite matrices

I need to find a positive semidefinite matrix $V$ such that $$\delta(V)\leq\beta$$ where $$\delta(V)=||I-\frac{1}{\mu}V^2||_F\quad \text{and }\quad\mu=\textbf{tr}(V^2).$$ Any ideas how can I get it? ...
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1answer
55 views

The true meaning of eigenvalues and eigenvectors, and positive (semi) definite matrix?

After studying some linear algebra, the true meaning of eigenvalues, eigenvectors, and positive definite matrix are still ambiguous. So, could you answer my questions or explanations about those ...
2
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1answer
51 views

Help with Dual problem in SDP

I'm having a problem to find the Dual of a Semidefinite programing problem: $$\min\;\;(tr(U)+tr(V))/2$$ $$s.t.\;\; \left[ \begin{array}{cc} U & X \\ X^T & V \end{array} \right]\succeq0$$ $$...
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3answers
28 views

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?
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56 views

Reformulate problem involving simultaneous diagonalization as linear program

(a) Let $\mathbb{S}^n$ be the set of $n\times n$ real symmetric matrices. Is minimizing $\lambda_{\max}(M)$ subject to $M\succeq 0$ and $\text{trace}(X_iM)=a_i$ for all $1\le i\le m$ a convex ...
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1answer
19 views

$LDL^\top$ for symmetric positive semidefinite matrices that are not positive definite

I have a symmetric positive semidefinite matrix (which is not positive definite) with integer entries and I know that I have an $LDL^\top$ decomposition for it (well mainly because Maple gives me one)....
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1answer
102 views

Trace inequality for a product of p.s.d. matrices and their pseudo inverse.

Let $A, B_i$ be positive semidefinite real matrices. Let $\dagger$ stand for the Moore-Penrose generalized inverse. I managed to prove that if $\operatorname{Ran}B_1\subseteq\operatorname{Ker}B_2$ ...
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1answer
30 views

Is a convex combination of positive semi-definite matrices itself positive semi-definite?

Suppose I have $\Sigma_1, \dots, \Sigma_n$ all positive semi-definite, and $\lambda_1, \dots, \lambda_n$ such that $\lambda_i \geqslant 0$ and $\lambda_1 + \dots + \lambda_n = 1$. Is it true that $\...
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1answer
50 views

Does $ 0 \leq A \leq B \implies 0\leq B^\dagger \leq A^\dagger$?

The question is stated in the title. Where $A$ and $B$ are positive definite real matrices and $\dagger$ stands for the Moore Penrose inverse. This question link1 seems to be close but it has an ...
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1answer
38 views

Is $\pmb{x}<\pmb{y}$ implies $A\pmb{x}<A\pmb{y}$ when $A$ is positive definite?

I think the title is self-explanatory. I would like to have a small proof supporting your statement. $\pmb{x}$ and $\pmb{y}$ denote vectors and $A$ denote a square positive definite matrix.
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26 views

Is a generalized inverse of a s.d.p matrix also s.d.p?

Let $A \in \mathbb{R}^{n\times n}$ be a simmetric semidefinite positive matrix. Let $A^+$ be a generalized inverse in the sense that $A A^+ A = A$. Is $A^+$ sdp? For the Moore Penrose inverse the ...
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3answers
47 views

Is $e^{-(L+L^T)}$ a positive definite matrix?

$L$ is a nonsymmetric Laplacian matrix and the zero eigenvalue is simple, is it true that $e^{-(L+L^T)}$ a positive definite matrix (I can be sure that it is non-negative matrix)?
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1answer
138 views

Directional derivatives of matrix trace functionals

Let $P$ be an $n\times n $ positive semidefinite matrix over $\mathbb{C}$, let $p\in\mathbb{R}$ be in the range $0<p<1$. Consider the function $g:[0,\infty)\rightarrow\mathbb{R}$ defined by $g(x)...
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1answer
26 views

Hessian and gradient in a matrix

There is following matrix: $$\begin{pmatrix}\nabla^2g(x) & \nabla g(x)\\\ \nabla g(x)^{T} & 1\end{pmatrix} \ge 0$$ the "$\ge$" is general inequality (not element-wise), meaning the matrix ...
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2answers
56 views

how to prove positive definite complex matrix

In order to prove that that a quadratic function is convex: $x^HAx$ , $A$ needs to be positive semi definite. Where $A= \Phi^H \Phi$ and $\Phi$ is a n×k matrix with possibly k linearly independent ...
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1answer
64 views

can this matrix be positive semidefinite?

Let $ S \in \mathbb{R}^n $ and $ A \in \mathbb{R}^{n \times n} $ Is it possible to find $ \alpha \in \mathbb{R} $, as a function of $A$, such that $$ S S^T ( \alpha 1_{n \times n} + A ) + ( \alpha 1_{...
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1answer
102 views

Closed form solution for block matrix A in $AXA^T=C$

Looking for help for closed form solution for A, given C and X in the matrix equation $AXA^T=C$ A (size $M x N$), X ($N x N$) and C ($M x M$) are (appropriately sized) block matrices $A = \...
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1answer
52 views

Does maximizing the Rayleigh quotient using the method of Lagrange multipliers require the matrix to be positive semidefinite?

I’m faced with the problem of maximizing a Rayleigh quotient: $$\max_h \,\, \frac{h^t H h}{||h||^2}$$ Which is equivalent to solving $$\max_h \,\, h^t H h $$ $$ s.t. ||h||^2=k >0, k \in \...