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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2answers
54 views

Why is this a smooth manifold? [duplicate]

I read that the space of quantum states, i.e. the space of density operators $\mathcal{S}_n = \{ \rho \in H_n : \rho \geq 0, \, \, Tr[\rho]=1\}$ is a smooth manifold of dimension $n^2-1$, without ...
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137 views

Analytical solution for a neat semidefinite program (SDP)

Let $A \in S^{n}_{+}$ be a positive semi-definite matrix with all entries being non-negative. I wonder if there is an analytical solution to the following SDP in correlation matrix $X \in S^{n}_{+}$ $$...
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1answer
14 views

$D_1 \Sigma D_1 \succeq D_2 \Sigma D_2$ with $\Sigma$ being positive semi-definite and $D_1 \succeq D_2$ diagonal matrices

Let $\Sigma \in \mathbb{R}^{n \times n}$ be a positive semi-definite matrix, and $D_1, D_2 \in \mathbb{R}^{n \times n}$ be two diagonal matrices with $(D_1)_{ii} \geq (D_2)_{ii} \geq 0$ for any ...
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22 views

Extremal positivity property of complete square

Consider the bilinear form $\displaystyle A(x,y) = \sum_{i, j} a_{i j} x_i y_j$ and biquadratic form $\displaystyle B(x,y) = \sum_{i, j, k, l} b_{i j k l} x_i y_j x_k y_l$ where $x, y \in \mathbb{R}^n$...
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37 views

Solve $A^T\Sigma A=I$ for $A$ with $\Sigma$ being psd

Suppose $\Sigma^{p\times p}$ is positive definite. For $A^{p\times p}$, the solutions to $A^T\Sigma A=I$ can be can be given by what? Suppose $\Sigma$ has eigendecomposition $\Sigma=PLP^T$. Then $P^T\...
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50 views

Importance of $\det(A^TA)$ in practical discussions

Let us assume that $A\in\mathbb{R}^{n\times n}$. It is known that since $A^TA$ is symmetric positive semi-definite, $\mathrm{det}(A^TA)$ is non-negative. I would like to know what is the importance of ...
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21 views

Inequality on determinant of product of positive semi definite matrices

Let $X$ and $Y$ be two $n\times n$ positive semi-definite matrices and $Z=X+Y$. Let $|X|$ be the determinant of $X$. Then can we say which one of $|I+Z|$ and $|(I+X)(I+Y)|$ is greater in magnitude?
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53 views

Is the negative normalized entropy convex?

The negative normalized entropy is defined as $$h:\mathbb{R}_{>0}^n \rightarrow \mathbb{R} \ , \ h(x)=\sum_{i=1}^n x_i\log \frac{x_i}{\sum_{j=1}^n x_j} \ .$$ Is this function convex? Its Hessian is ...
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1answer
36 views

Proving the difference between two matrices is positive definite

I'm working on the convex relaxation of a problem, and I came across the following question. Suppose I have a vector $x \in \mathbb{R}^n$ where $-1 \leq x_i \leq 1$ and a matrix $X$ whose diagonal ...
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1answer
25 views

Positivity of a given matrix

Let $A\in M_{n\times n}(\mathbb{R})$ be a real symmetric matrix and $H\in M_{n\times n}(\mathbb{C})$ be an invertible hermitian matrix. We are given that $A+H$ and $A-H$ are positive semi-definite. ...
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48 views

Given $A \in \mathbb{R}^{n \times n}$ such that $y^T A y \geq 0$ for any $y \in \mathbb{R}^n$, does $x^TAx=0$ imply $Ax=0$?

Given $A \in \mathbb{R}^{n \times n}$ such that $y^T A y \geq 0$ for any $y \in \mathbb{R}^n$, does $x^TAx=0$ imply $Ax=0$? I think i have proven it in the case where $A$ is symmetric (so $A$ is ...
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37 views

Optimisation of nonlinear matrix objective with semidefinite constraints (classical Fisher information)

Given the invertible symmetric block diagonal matrices $S = S_1\oplus S_2$ and $dS = dS_1 \oplus dS_2$, I want to find the symmetric matrix X with blocks $$ X = \left[ \begin{array}[cc] ~X_1 & X_{...
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25 views

Under what condition can a symmetric matrix becomes PSD

I've just learned the concept of PSD in class, and I've got a little confusion. If A is a symmetric matrix represented by $\pmatrix{a&b\\b&c}$, what conditions should be satisfied to becomes ...
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1answer
58 views

Why the set of all $n{×}n$ PSD matrices is not a subspace of all $n{×}n$ symmetric matrices?

I know this question is kind of basic, but I am pretty confused about that. PSD is also symmetric, how can it not belongs to the symmetric's subspace? I have no idea to prove it.
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1answer
26 views

If $B^{m\times m}$ is positive semidefinite, is $A^TBA$ is positive semidefinite?

If $B^{m\times m}$ is positive semidefinite, is it true that $A^TBA$ is positive semidefinite with $A^{m\times n}$? I think it is, because all of my counterexamples have failed, but I don't know how ...
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1answer
16 views

Revisit “What is the rank of $Q+Q^T-Q\circ I$ if $Q = qq^T$”

Based on my previous problem: What is the rank of $Q+Q^T-Q\circ I$ if $Q = qq^T$ $Q = qq^T$, $q\in \mathbb{R}^n$. Suppose $q^Tq=1$. $A\circ B:$ Hadmard (elementwise) product $Q\circ I$ just takes the ...
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61 views

Show that $\Sigma$ is positive semidefinite using Jensen's inequality

Let $X\in \mathbb R^d$ be a random vector and $\mu=E(X)\in \mathbb R^d$ be its mean. Let $\Sigma=E[(X-\mu)(X-\mu)^T]\in\mathbb R^{d\times d}$ be the variance-covariance matrix of X. Show that $\Sigma$ ...
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40 views

$A^{-1} - M^{-1}$ symmetric positive semidefinite

$A$ is a symmetric positive definite matrix and $ X^{ \top} X = I - A^{1/2} \overline{M}^{-1} A^{1/2}$. We already know that $\overline{M}^{-1}$ is symmetric positive definite. Now it's about to show ...
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1answer
15 views

Positive semidefinite inequality under tensor product

Let $\rho, \sigma$ be finite dimensional positive semidefinite matrices with trace less than or equal to 1. Let $\lambda(1)$ be the smallest nonnegative number (possibly infinity) such that $$\rho \...
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1answer
41 views

Showing that a general Hessian matrix is positive semidefinite

Given vector $a \in \mathbb{R}^n$, show that the scalar field $g : \mathbb{R}^n \to \mathbb{R}$ defined by $$g(\mathbf{x}) = -\log ( f(\mathbf{x}))$$ where $$f(\mathbf{x}) = \dfrac{1}{1+\exp(-a^T\...
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2answers
69 views

Positive-semidefiniteness of combination of two positive-semidefinite matrices

I am struggling for a while in proving the positive-semidefiniteness ( in the usual sense, i.e., $A$ is Hermitian and $x^*Ax\geq 0\,$ for all $\,x \in \mathbb{C}^2\,\big)$ of the following matrix $$A :...
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1answer
82 views

A positive semidefinite matrix multiplied by any matrix and its transpose is positive semidefinite?

Let $P$ be a positive semidefinite matrix and $A$ any matrix. Is $X=APA^\intercal$ positive semidefinite, if so give a proof, if not a counterexample. (not sure if the context of the problem means ...
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1answer
28 views

A hermitian positive semi-definite matrix with all entries on the complex unit circle has rank one

Can someone think of an easy way to prove (or disprove) the above conjecture? For $2\times 2$ matrices, the conjecture is trivially true. For $3\times 3$ matrices, the non-negativity of the ...
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1answer
27 views

The equivalence of a 2 by 2 positive semidefinite matrix and a 3 by 3 positive semidefinite matrix?

I came across the following: $$\begin{bmatrix} -x^TAx-2b^Tx+c &-(Ax+b)^TR\\ -R(Ax+b) & \lambda I -RAR \end{bmatrix}\geq0 \iff \begin{bmatrix} b^TA^{-1}b+c & 0 &(x+A^{-1}b)^T\\ 0 & ...
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The question on the “=” condition on $(x'By)^2\leq x'Ax y'Cy$

Let $M=\begin{pmatrix}A&B\\B'&C\end{pmatrix}$ be semi-positive-definite, that is, $X'MX\geq 0$. Here $A$ is $m\times m$, $C$ is $n\times n$. Then by checking with $X=\begin{pmatrix}x\\ty\end{...
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51 views

What is the adjoint operator for the Diag function?

The Diag function sends a vector x in R^n to the diagonal nxn matrix with the components of x along the diagonal. The Diag function is a linear map from R^n to S^n_+, the set of symmetric positive ...
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1answer
18 views

Find an equivalent condition to $\operatorname{tr}(A^{\ast}B)=0$ for two complex hermitian matrices $A, B$ with rank of $B$ 1.

I have two matrices $A,B$ given that both are positive semi-definite, complex. Another given is that $B$ has rank $1$ and is hermitian. Additionally $B=ww^{\ast}$ where $w\in \mathbb{C}^{m}$ with $w_k=...
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17 views

Is the function of a positive (semi) definite matrix convex?

If you have a matrix of variables say $X=AA^{T}$ which by definition must be either positive definite or positive semidefinite, and you have some function $f(X)$, as $X \in C$, where $C$ is a cone, ...
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40 views

Upper/lower bounds for Frobenius inner product of matrices

Suppose I have $$-1 \leq \langle D,qq^T\rangle \leq \cos\theta \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$ $D\in \mathcal{S}^{4\times4}$, symmetric and traceless matrix (not knowing ...
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27 views

Set of solutions for given inequality

Given the matrix $A\in\mathbb{R}^{n\times n}$ with all eigenvalues inside the unit circle and the symmetric positive definite matrix $P\in\mathbb{R}^{n\times n}$ satisfying $ APA^\top-P+I=0 $, I need ...
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1answer
34 views

Is A-B positive semidefinite if A's eigenvalues are greater than B's

I know that if matrices $A$ and $B$ are positive semi-definite and $A\succeq B$ ($A-B$ is positive semi-definite), then the $i$-th eigenvalue of $A$ is greater than or equal to the i-th eigenvalue of $...
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42 views

When does a log-convex function not form an positive semi-definite matrix?

If we consider the function $f(x)$ and the matrix A formed as follows $$ A_{ij} = f(\frac{x_i + x_j}{2}) $$ and state that $f(x)$ is log-convex and non-negative $$ f(x_1)^tf(x_2)^{(1-t)} \geq f(tx_1 + ...
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1answer
54 views

How to prove that all the eigenvalues of the matrix satisfy $0\leq\lambda\leq1$?

Assume that $D\in{\bf R}^{n\times n}$ is a symmetric positive definite matrix, $B\in{\bf R}^{m\times n}$ is an arbitrary matrix, and $C\in{\bf R}^{m\times m}$ is a symmetric positive semidefinite ...
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1answer
24 views

Generalising positive semi-definiteneness of 2×2 to N×N matrix

If I have some function $G(x), \quad G:X\to\mathbb{R}$, which is used to construct a matrix as follows $$ A_{ij} = G\Bigl(\frac{x_i + x_j}{2}\Bigr) $$ and I can show that the $2\times2$ matrix has all ...
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19 views

Partial eigendecomposition of a positive semi-definite matrix

Any positive semi-definite matrix $A$ can be decomposed into $$ A = Q \Lambda Q^\dagger, $$ where $Q$ is a unitary matrix and $\Lambda$ is a diagonal matrix. Now I would assume that there is a ...
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Optimization with matrix's inverse in inequality constraint.

my optimization problem may be difficult to correspond to a known optimization problem. Here it goes: Assume there is an unknown vector $x$, whose entries are all in the range $[0, 1]$. The ...
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1answer
78 views

Positive Semidefiniteness of a Hermitian Matrix

I came across this problem and I'm not sure if the proof I have found is OK. I may be overcomplicating things. Any suggestions or confirmation of correctness would be appreciated. The Problem: Let $A$ ...
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2answers
58 views

If the trace of the product of two matrices is zero,is the product of the matrices zero?

Does $\:\operatorname{ Tr}(XY) = 0 \Rightarrow XY=O $, for all $X,Y \succeq O$ ?
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1answer
37 views

Efficient product of vector with inverse of positive definite circulant matrix plus real diagonal matrix?

Question: Hi! Would anyone happen to know an efficient way to compute: $$ u^\top [C + D]^{-1}, $$ where: $C$ is a square matrix, and is real, positive definite, and circulant $C$ is too large to fit ...
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1answer
21 views

In a Clifford algebra over an $n$-dimensional positive-semidefinite space, is a product of any number of vectors a product of at most $n$ vectors?

This is related to the Cartan-Dieudonne theorem. In a Clifford algebra over an $n$-dimensional real vector space $V$ with quadratic form $a\cdot a\geq0$ for all $a\in V$, can any product of vectors $...
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115 views

Is there a name for “positive-semidefinite” quadratic forms when the base field is not $\mathbb R$?

Consider a vector space $V$, with a symmetric bilinear form $\cdot$ , over a field $\mathbb F$ with characteristic not $2$. Suppose, for any $x\in V$, if $x\cdot x=0$, then $x=0$. What is this ...
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70 views

Show that the following matrix is positive semidefinite

Problem: Let $0\leq a_1 \leq a_2 \leq \dots \leq a_n < \infty$. Show that the $n\times n$ matrix \begin{align*} A = \begin{pmatrix} a_1 & a_1 & \dots & a_1\\ a_1 & a_2 & \dots &...
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63 views

How to prove that the element-wise exponential of a symmetric matrix is not always positive-definite? [duplicate]

The matrix exponential $e^X$ of a square symmetric matrix $X$ is always positive-definite, where $$ e^X = \sum_{k=0}^\infty \frac{1}{k!} X^k $$ Does the exponentiation of each element of the symmetric ...
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1answer
51 views

How to bound the norm of this matrix?

Let $(A_n)$, $(B_n)$, $(C_n)$ be sequences of $n\times k$ matrices, where $k<n$ is fixed and $$A'_nA_n=B'_nB_n=C'_nC_n=I_k \hspace{1cm} $$ $$A_n'C_n=0$$ $$\|A_n\|_F=\|B_n\|_F=\|C_n\|_F=\sqrt{k}$$ ...
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1answer
24 views

operator 2 norm and symmetric positive semidefinite matrix

Let $A$ be a symmetric positive semidefinite matrix and $u$ be a unit vector. Then $u^TAu \leq \|A\|_2$ ? Here $\Vert \cdot \Vert_2$ is the induced/operator 2-norm defined as $\| A \|_2 = \sup \limits ...
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1answer
33 views

Proving $A^T B A$ is positive definite or semidefinite, when $B$ is positive definite

Let's assume we have a positive definite matrix called $B$. (Here I assume the definition of the positive definite matrix stating $B$ should be symmetric.) How could we conclude that $A^T B A$ is ...
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36 views

Criteria for positive-definiteness of higher-order tensors

A second-order tensor is a bilinear form $$\tag{1} B(x, y)=\sum_{i, j=1}^d b_{ij}x_iy_j, \qquad \text{where }x, y\in \mathbb R^d,$$ and we will always assume without loss of generality that it is ...
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1answer
60 views

Proof Verification of a Linear Algebra problem

Consider a symmetric and positive semidefinite matrix $A \in \mathbb{R}^{n \times n}$ Assume additionally that there exists no vector $x \in \mathbb{R}^n \setminus \{ 0 \}$ which satisfies $x^T A y = ...
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1answer
27 views

Maximize a quadratic convex function with a symmetric structure

Let $S = \begin{pmatrix} A & B \\ B & A \end{pmatrix} \in \mathbb{R}^{2n \times 2n}$ be a positive semi-definite matrix with matrices $A, B \in \mathbb{R}^{n \times n}$ being symmetric. Let $...
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1answer
66 views

Is every positive definite matrix also positive semidefinite?

I am in trouble with the definitions of positive definite and positive semidefinite matrices. By definition, does the following implication hold? $$\mbox{positive definite} \implies \mbox{positive ...

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