# Questions tagged [positive-semidefinite]

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

655 questions
Filter by
Sorted by
Tagged with
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?
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 ...
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: ...
43 views

10 views

### Positive semidefinite sequence, estimators of the acvs

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

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 ...
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 ...
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'...
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 ...
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 ...
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 ...
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} \;\...
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 ...
43 views

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 ...
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 ...
61 views

I am trying to understand whether the following expression is a convex function: $$f\left (\mathbf{X} \right ) = \mathrm{tr}\left( \left ( \mathbf{\Lambda}+\alpha_0 \mathbf{\Psi}^T \left( I-e^\... 1answer 30 views ### Is tensor power an operator monotone function? Let A,B be postitive semi-definite operators on a finite dimensional Hilbert space. Is the following true? \begin{equation} A \geq B \ \Rightarrow A^{\otimes n} \geq B^{\otimes n} \quad n=1,2,3,... \... 1answer 111 views ### Positive-definiteness of a matrix with entries \frac1{(a_i+a_j)^\alpha} Let 0<a_1<\ldots<a_n be real numbers, and let \alpha>0 be given. Consider the matrix A=\begin{pmatrix}\frac1{(a_i+a_j)^\alpha}\end{pmatrix}_{1\leq i,j\leq n}. Then is A positive-... 1answer 67 views ### Minimize the Trace of 2 PSD Matrices Product Subject to a Constraint on the Trace Given  A \in \mathcal{S}_{+}^{n \times n}  (PSD Matrix) with  \lambda_{max} \left( A \right) < 1  solve the following optimization problem:$$ \arg \min_{X \in \mathcal{S}_{+}^{n \times n}} \...
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{...