Questions tagged [positive-semidefinite]

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

Filter by
Sorted by
Tagged with
1
vote
1answer
17 views

Would symmetry positive semi-definite matrix always decomposable?

Given symmetry positive semi-definite matrix $A \in R^{n\times n}$. And $Det(A) \geq 0$. Would there always exist real matrix $B$, such that $A = B \cdot B^T$? If so why? Or why not?
2
votes
2answers
34 views

The norm $\|S-Q\|_F$ where $Q$ is orthogonal is minimised by $Q=I$

Problem: Suppose that $S$ is symmetric and semi-positive-definite. Let $\|\cdot \|_F$ be the Frobenius norm. Show that $$\|S-I \|_F \leq \|S-Q\|_F$$ for all orthogonal matrices $Q$, where $I$ is ...
0
votes
0answers
18 views

Show that there exist matrices $A$ and $B$ such that this matrix series is not positive semi-definite

This is a follow-up on a question I asked before. Suppose that $A$ is a $k \times k$ matrix with $\|A\|_2 \leq 1$ and $B$ is a $k \times k$ positive semi-definite matrix. Consider the following sum: ...
0
votes
1answer
43 views

Proof of positive semi-definiteness of hat matrix product

Consider the matrices $B = \begin{bmatrix}1 &0\\-1&1\\0&1\end{bmatrix}$, $Y = \begin{bmatrix}x_1&0 & 0\\0&x_2&0\\0&0&x_3 \end{bmatrix}$, $\tilde{B} = YB$ and $T =...
0
votes
0answers
17 views

Mean orthogonal projection on a vector of multidimensional normal distribution

Let $X\in\mathbb{R}^n$ be a normally distributed random vector with mean zero and positive definite covariance matrix $\Sigma$, i.e. $X\sim\mathcal{N}(0,\Sigma)$. Treating $X$ as a column vector, the ...
0
votes
0answers
19 views

Converting semidefinite program into standard form linear program

I'm new to semidefinite programming and I'm having trouble computing what I think is a fairly simple semidefinite linear program. I need to maximize $$\mathrm{Tr}[H X]$$ subject to some linear ...
2
votes
1answer
30 views

Show that this matrix series is positive semi-definite

$A = B^\top B$ is a $k \times k$ symmetric positive semi-definite matrix and $\|A\|_2 \leq 1$. Consider the following sum for $c$: $$S_n = \sum_{i=0}^{n-1} (n-i) A^{i} - \frac{1}{c} \sum_{i=0}^n A^{i}...
1
vote
1answer
24 views

Squared exponential kernel with Manhatten distance does not result in positive semi-definite matrix

Here it is stated that the squared exponential covariance function $$C(d) = e^{-(\frac{d}{V})^2},$$ where $V$ is a scaling parameter and $d$ is a distance between two points, is a stationary ...
1
vote
2answers
29 views

Construct a diagonal-1 gram matrix with given eigenvalues

I got a question concerning positive semidefinite matrices. Given a positive semidefinite matrix $M\in \mathbb{R}^{n\times n}$, we know that its trace $\text{tr}(M)$ equals to the sum of its ...
1
vote
2answers
34 views

In general, are functions of this form convex?

I know that quadratic functions, i.e. functions of the form $f(x) = \frac{1}{2}x^TAx+b^Tx+c$ (with $A\in\mathbb{R}^{nxn}$, $b\in\mathbb{R}^n$, $c\in\mathbb{R}$), are convex over $\mathbb{R}^n$ when A ...
0
votes
0answers
23 views

Hessian matrix (2x2) has trace,det=0 but a negative eigenvalue--is this matrix PSD?

I'm trying to show whether the function $f(x)=sin(\frac{\pi x_2} {2})$ is convex for $x\in \mathbb{R}^2$, where $0<x_1\le2$ and $0<x_2\le1$ When I compute the Hessian, I find $$H=\nabla^2f=\...
0
votes
0answers
14 views

Given that a matrix is PSD, can we show whether this function is strictly convex?

For this problem it's given that $Q\in \mathbb{R}^{n\times n}$ is positive semi-definite, and now I'm trying to figure out whether $$g(x) = \sqrt{x^TQx-1}$$ is convex over $\mathbb{R}^n$ I did a ...
0
votes
0answers
24 views

Is $\min\limits_{x,y}\{\sqrt{1+x^2}-\sqrt{1-y^2}\mid C\}$ is convex problem?

I need to understand if $$\min_{x,y}\left\{\sqrt{1+x^2}-\sqrt{1-y^2}\,\middle|\,\frac{\sqrt{(x-1)^{2}+y^2}}{\sqrt{x^2+(y-2)^2}}\leq\frac12,\ |y|\leq1\right\}$$is convex problem. I found that the ...
0
votes
1answer
18 views

Necessary and sufficient condition for the existence of a positive operator in a finite-dimensional inner product space

Exercise from PR Halmos's Finite-Dimensional Vector Spaces, Edition 2 (Ex 8 on page 142). If $x$ and $y$ are nonzero vectors (in a finite-dimensional inner product space) then a necessary and ...
1
vote
1answer
19 views

Semi-positivity of an Hermitian matrix

I´m triying to show that an $n\times n$ matrix $M$ such that $M_{ij}=e^{\gamma_{ij}}$ where $\gamma_{ji}=\gamma_{ij}^{\ast}$ are complex numbers is a positive semi-definite matrix. I made a proof of ...
0
votes
1answer
27 views

Eigenvectors of a sum of PSD matrices

Lets say I have two real-valued positive-semidefinite matrices $A$ and $B$ with eigenvectors given by $v_i$ and $w_i$, respectively. Does this tell me anything about the eigenvectors of $C = A + B$ I ...
3
votes
2answers
68 views

If a matrix is positive-semidefinite, is Hermitian, and has a trace of 1, is it idempotent?

I have a matrix $A^{n\times n}$ that is positive semi-def. and Hermitian. Also, $Tr(A) = 1$. I want to show that $A$ is idempotent. Are these properties enough? If so, how would one show this? If not,...
0
votes
3answers
24 views

Show that this function is convex given that Q is positive semi-definite

For this problem it's given that $Q\in \mathbb{R}^{n\times n}$ is positive semi-definite, and now I'm trying to show that $$f(x) = \sqrt{x^TQx+1}$$ is convex over $\mathbb{R}^n_{++}$ I've tried to ...
0
votes
1answer
21 views

Let $A$ be positive semi-definite. Show that $a_{i,i}a_{j,j} \geq |a_{i,j}|^2$, $1\leq i<j \leq n$

I was trying to do it by induction. First let $$ A = \begin{bmatrix} a & b \\ \overline{b} & c \end{bmatrix}.$$ Since $A$ is PSD, $det(A) = ac - |b|^2 \geq 0$. So $ac \geq |b|^2$. Now ...
0
votes
1answer
38 views

Eigenvalues of matrix product $ADA^T$

I am considering such a matrix product, namely $ADA^T$, where $A$ is an $n\times m$ matrix with $n>m$, $D$ is an $m\times m$ positive diagonal matrix. I understand the matrix product is positive ...
2
votes
0answers
76 views

Would such a matrix exist?

Could anyone help me search for a positive semidefinite matrix $\left(a_{i,j}\right)_{4\times4}$of rank 3 with $a_{i,i}=3$ and its all 3 by 3 principal minor matrices having minimal eigenvalue $\...
0
votes
0answers
10 views

Positive semidefinite sequence, estimators of the acvs

definition of positive semidefinite for a sequence $s_n$: $\forall t_1, \cdots, t_n \in \mathbb T, \forall a_1, \cdots, a_n \in \mathbb R^*\colon$ $$ \sum_1^n \sum_k^n s_{t_j-t_k} a_j a_k \geq 0 $$ ...
0
votes
1answer
27 views

Difference between two positive semidefinite matrices

Let $A, B \in \mathbb{R}^{n \times n}$ be symmetric positive semidefinite matrices and: \begin{equation} \lambda_{max} (A) \leq \lambda_{max} (B) \end{equation} Can we prove that the matrix $C = A - ...
0
votes
1answer
30 views

Question about the diagonalizability

In some textbooks it is mentioned that, if a matrix is Positive Semidefinite, then all its diagonal entries are nonnegative and all the principal submatrices of PSD obtained by removing any number of ...
1
vote
0answers
20 views

How to calculate $\max\{\frac{1}{2}\sum_{i=1}^5(1-X_{i,i+1}) \mid X \text{ PSD}, X_{ii}=1 \forall i\in [5]\}$ analytically.

I'm trying to calculate $sdp(C_5) =\max\{\frac{1}{2}\sum_{i=1}^5(1-X_{i,i+1}) \mid X \text{ PSD}, X_{ii}=1 \forall i\in [5]\}$ analytically. I've already bounded it: As $\begin{pmatrix}1 & -\...
0
votes
2answers
47 views

Can a symmetric positive semi-definite matrix be transformed to any symmetric positive semi-definite matrix with the same rank?

Given a symmetric positive semi-definite matrix $A\in\mathbb{R}^{n\times n}$ and denote by $rank(A)$ the rank of $A$. Let $\bar A=Q^TAQ$, where $Q\in\mathbb{R}^{n\times n}$ is an arbitrary invertible ...
0
votes
0answers
12 views

A globally positive semi-definite second-order polynomial can be written as a completing square?

Consider the following second-order polynomial \begin{equation} f(y,z)=k_1+k_2\,y+k_3\,z+k_4\,y^2+k_5\,y\,z+k_6\,z^2, \end{equation} where $k_i\in\mathbb{R}$ $(i=1,\cdots,6)$ are constants. Suppose ...
1
vote
2answers
169 views

A and B are real, symmetric and positive semi-definite matrices of the same order; is AB diagonalizable?

I've already proved this with B positive definite using the fact that $B^{-\frac{1}{2}}$ does exist and AB is similar to $A^{\frac{1}{2}}BA^{\frac{1}{2}}$, but since B is positive semi-definite I don'...
0
votes
1answer
49 views

Show that, if $A$ is PSD, then there is an optimal solution $x_i, y_j$ of the below given optimization problem such that $x_i = y_i$ for all $i$

I'm trying to prove that for the optimization problem $$\max \{\sum_{i,j=1}^n A_{ij}<x_i,y_j> \mid x_1,\dots x_n,y_1,\dots y_n\in S^{2n-1}\}$$ that if $A$ is PSD, then there exists an optimal ...
0
votes
2answers
19 views

Positive definite implies positive semi-definite

I have read in the post, but there is something missing I'd like to know. I do believe it's not a duplicate post. I have a specified question, probably a trivial one. First, let $(s_n)_{n\geq 0}$ be ...
2
votes
4answers
95 views

Variation of Least Squares with Symmetric Positive Semi Definite (PSD) Constraint

I am trying to solve the following convex optimization problem: \begin{align} & \min_{W} && \sum_{i=1}^n (\mathbf{x}_{i}^TW\mathbf{x}_{i} - y_i)^2 \\\\ & s.t. && W \succcurlyeq ...
0
votes
1answer
27 views

A matrix optimization problem that resembles a standard semidefinite program

I have a constrained matrix optimization problem as follows \begin{align} \max\limits_{X} \;\; &\mbox{tr}\Big( \left(C - \frac{1}{2} B R^{-1} S \right) \Lambda X^T \Big) \\ \text{subject to} \;\...
0
votes
0answers
11 views

How to prove that an inner matrix of LU decomposition of PD matrix is also PD?

A is symmetric positive definite. Let A=L*U (lU decomposition of the matrix A) We need to prove that B is also positive definite. Her's what I did: Until now I have proved that the sum of all ...
1
vote
1answer
43 views

Convex optimization under a positive semidefinite constraint involving projections

I am interested in finding the smallest operator $X$ in the Frobenius norm (also called the Hilbert-Schmidt norm) $$\begin{array}{ll} \text{minimize:} & \lVert X \rVert_F^2\\ \text{subject to:} ...
0
votes
1answer
21 views

Proving the difference of two matrices with similar properties is psd

I have a vector $w \in \mathbb{R}^n$ and a symmetric matrix $V \in \mathbb{S}^{n}$. I have the following properties: $w_i \geq 0$ for all $i=1,\ldots,n$ $\sum \limits_{i=1}^n w_i = 1$ $V_{i,j} \geq ...
0
votes
0answers
16 views

Is it possible to construct a PSD matrix that has multiple null vector rows?

I know that if I generate a random matrix $\mathbf{A}\in\mathbb{C}^{n\times n}$ in MATLAB, then $\mathbf{B} = \mathbf{A}\mathbf{A}^{\dagger}\in\mathbb{C}^{n\times n}$ is a Hermitian (i.e., $\mathbf{B}^...
0
votes
1answer
16 views

matrix psd inequality, for addition

Given four matrices $A, \widetilde{A}, B, \widetilde{B} \in \mathbb{R}^{n\times d}$, if $A^{\top} A \approx_{\epsilon} \widetilde{A}^{\top} \widetilde{A}$, $B^{\top} B \approx_{\epsilon} \...
0
votes
0answers
43 views

Prove that $A$ is positive semidefinite iff $\alpha \geq 1$

Given a symmetric matrix $n\times n$ $$A_{i,j} = \begin{cases} \alpha & \text{ if } i=j\\ 1 & \text{ if } i \neq j \end{cases}$$ Prove that $A$ is positive semidefinite iff $\...
1
vote
0answers
14 views

What does edge-transitive imply for the adjacency matrix of graph?

I'm trying to prove that if a matrix $G$ is edge transitive, we can say that $$\vartheta(G) = \frac{n\lambda_{min}(A_G)}{\lambda_{max}(A_G) -\lambda_{min}(A_G)}$$ Where $\vartheta(G)$ is defined in ...
0
votes
0answers
11 views

compare variances of ols regression function

Let $\tilde{f}$ denote the ols estimation of $f(x) = \beta_0 + \beta_1x_i$. Let $\hat{f}$ denote the ols estimation of $f(x) = \beta_0+\beta_1x_i+\beta_2x_i^2$. I wish to compare the variances of $\...
1
vote
0answers
42 views

Why is the theta number / Lovasz number additive on disjoint graphs?

Let $G=(V,E)$ be the disjoint union of two graphs$G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$. That is, $V_1$ and $V_2$, $V=V_1 \cup V_2$ and $E=E_1\cup E_2$. We want to show that $$\theta(G) = \theta(G_1) + \...
1
vote
0answers
12 views

Is there an identity for the determinant of the sum of positive definite or semidefinite matrices?

I am familiar with the Minkowski inequality, but that is only for two matrices. What about three? four? Or in general, the determinant of the sum of matrices. Specifically positive definite or ...
0
votes
1answer
35 views

Is Positive Semi-Definiteness of a Matrix a loose measure of Independence?

I am trying to understand D-optimality criteria which tends to minimize the covariance matrix between two random variables. Now keeping that aside and in general, having PSD of a covariance matrix ...
1
vote
2answers
61 views

Moore-Penrose pseudoinverse: product on left with another matrix

The following problem comes from studying the conditional expectation of a multivariate normal distribution. Let $n\ge2$ be an integer, and let $\Sigma$ be a positive semidefinite, symmetric $n\times ...
0
votes
0answers
51 views

Difference of square roots of positive semidefinite matrices

Let $Y$ and $X$ be positive semidefinite matrices of size $d \times d$. We define $\sqrt{Y}$ and $\sqrt{X}$ as the unique symmetric positive semidefinite matrices that satisfy $(\sqrt{Y})^2 = Y$ and $(...
0
votes
0answers
52 views

Is the trace of the matrix exponential convex or non-convex?

I am trying to understand whether the following expression is a convex function: $$ f\left (\mathbf{X} \right ) = \mathrm{tr}\left( \left ( \mathbf{\Lambda}+\alpha_0 \mathbf{\Psi}^T \left( I-e^\...
2
votes
1answer
30 views

Is tensor power an operator monotone function?

Let $A,B$ be postitive semi-definite operators on a finite dimensional Hilbert space. Is the following true? \begin{equation} A \geq B \ \Rightarrow A^{\otimes n} \geq B^{\otimes n} \quad n=1,2,3,... \...
1
vote
1answer
111 views

Positive-definiteness of a matrix with entries $\frac1{(a_i+a_j)^\alpha}$

Let $0<a_1<\ldots<a_n$ be real numbers, and let $\alpha>0$ be given. Consider the matrix $A=\begin{pmatrix}\frac1{(a_i+a_j)^\alpha}\end{pmatrix}_{1\leq i,j\leq n}$. Then is $A$ positive-...
0
votes
1answer
67 views

Minimize the Trace of 2 PSD Matrices Product Subject to a Constraint on the Trace

Given $ A \in \mathcal{S}_{+}^{n \times n} $ (PSD Matrix) with $ \lambda_{max} \left( A \right) < 1 $ solve the following optimization problem: $$ \arg \min_{X \in \mathcal{S}_{+}^{n \times n}} \...
0
votes
0answers
26 views

Completing matrix products to PSD matrices

Suppose matrices $A \in \mathbb{R}^{r \times n}$ and $\Sigma_1 \in \mathbb{R}^{n \times n}$ are given s.t $\Sigma_1$ is a rank-1 PSD matrix. Are there non-trivial choices of a matrix $B \in \mathbb{...

1
2 3 4 5
14