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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Are all entrywise nonnegative positive semidefinite matrices has a rank-1 decomposition with nonnegative vectors?

I know that all positive semidefinite matrices has a rank-1 decomposition. (Equivalently, all quadratic nonnegative polynomial is sum of squares of linear function.) $$A = \sum_{i=1}^r x_i x_i^T = X X^...
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Conditions on 3 x 3 matrix elements for the matrix to be positive semi-definite by strictly using the definition of positive semi-definite matrices.

A matrix $M$ is considered positive semi-definite if $$ z^{T}Mz \geq 0 $$ Using this definition, I am trying to discover all conditions required on the elements of the 3x3 symmetric matrix $M$ for it ...
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Euclidean projection on convex set of positive semidefinite matrices

Define the Euclidean projection for a convex set $C$ as follows $$\pi_C(y) := \min_{x \in C} \| y - x \|_2^2$$ How would we find the projection map when $C$ is the cone of positive semidefinite ...
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What conditions does positive semidefiniteness impose on the matrix elements?

Let $A$ be an Hermitian $2\times 2$ complex matrix. We can always write it as $$A =\begin{pmatrix}a & \alpha \\ \bar\alpha & b\end{pmatrix}$$ for some $a,b\in\mathbb{R}$ and $\alpha\in\mathbb{...
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Inequalities on positive semi-definite matrices

I encountered the following question from Problems and Solutions in Introductory and Advanced Matrix Calculus by Steeb and Hardy (P.39 Problem 3). Question. Let $A\in M_n(\mathbb{C})$ be a positive ...
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Find matrix with given properties

I proved that for a positive semidefinite matrix holds: $$ x^TAx =0 \Rightarrow Ax=0. $$ I wanted to come up with an counterexample for an indefinite say $2 \times 2$ matrix: So there is a $x$, such ...
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Estimates with square root of sum of positive semi definite matrix and diagonal matrix

Let's say i have a $n \times n$ positive semi definite matrix $A$ with real entries, $V$ is a vector in $\mathbb{R}^n$ and $\lambda>0$ such that $\lambda |V|^2 \leq a$ for $a>0$. My text says ...
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How to check whether this matrix is positive semidefinite?

Given the following symbolic matrix \begin{equation*} A= \begin{pmatrix} -\cos(\theta_1-\theta_2)&\cos(\theta_1-\theta_2)&0 \\ \cos(\theta_1-\theta_2) & -\cos(\theta_1-\theta_2)-\cos(\...
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Sufficient conditions for an indefinite real symmetric matrix

Given a real symmetric matrix $S$ of order $n$, assuming $\mathbf{x} \equiv \mathbf{e}_i$ with $i \in [1,\,n]$ we have: $$ \mathbf{x}^t\,S\,\mathbf{x} = s_{i,i} $$ so if $s_{i,i}=0$ for at least one $...
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Graph Laplacian for weight matrix with negative edges

How can I normalize my weight matrix to get a positive semi-definite Laplacian, if I am using a weight matrix with negative edges?
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Give an example of a symmetric matrix $A$ such that $A\in S_4^+$ but $A\neq \sum_{k=1}^NX_k\otimes Y_k$ where both $X_k,Y_k\in S_2^+$.

I have trouble in finding an example of a symmetric matrix $A\in M_4(\mathbb{R})$ such that $A$ is positive semidefinite matrix but $A$ cannot be written as sum of tensor product of matrices $X_k,...
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What can cause the Hessian to lose positive semi-definiteness in Newton's method

I have been looking at some code for Newton's method. My sources (Nocedal and Wright), indicate that Newton's method works very well in many cases, but that often the method can fail because the ...
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Definiteness of Matrix over Torus

I am interested in characterizing the definiteness of the following bilinear form over the torus: $$\langle x, A x \rangle$$ $$A = A^T, A \in \mathbb{R}^{2n \times 2n}$$ Where the inner product ...
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Polar decomposition of a linear combination of unitary matrices

Consider a complex-valued square matrix $M$ of the form $$M = \frac{1}{2}\left(U_1 + e^{-i\phi}U_2\right),$$ where $U_1$ and $U_2$ are unitary matrices and $\phi$ is a real number. Moreover, consider ...
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Concavity of smallest positive eigenvalue of a semidefinite matrix

Let $A\in\mathbb{R}^{d\times d}$ be positive semidefinite with rank $n<d$. Hence $A$ is not positive definite. Is the function $$ A \mapsto \lambda^{+}_{\text{min}}(A)$$ concave? Here $\lambda^{+}_{...
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Image of a $3 \times 3$ PSD cone under a linear transformation

Let $X \in \mathcal{S}_+^3$ be a $3 \times 3$ positive semidefinite matrix, and let $\mathcal{A}$ be a linear transformation from $\mathcal{S}^3 \rightarrow \mathbb{R}^3$ defined as follows: \begin{...
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Bounding norm of a vector

The problem Consider two $d$-dimensional POVMs, $ \mathcal{E}$ and $\mathcal{F}$, given by POVM elements $\left\{E_{1}, \ldots E_{n}\right\}$ and $\left\{F_{1}, \ldots F_{n}\right\}$, respectively. (...
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Eigenvalue of PSD matrices by constrained SDP program

Given Lemma 1, I want to prove the following corollary. lemma 1 (Rayleigh Quotient): Given matrix $A \succeq 0$, \begin{equation} \lambda_{\min} (A) = \min_{x \in \mathbb R^n}\frac{x^\top A x}{x^\...
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How to prove a result dealing with Loewner order when one of the matrices is only positive semidefinite?

Suppose $\mathbf{A}\in\mathbb{C}^{M \times M}$, and diagonal matrix $\mathbf{B}\in\mathbb{C}^{M \times M}$. $\mathbf{A}\succeq \mathbf{0}$, and $\mathbf{0}\preceq\mathbf{B}\preceq\mathbf{I}$. How to ...
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Suppose $X \in [-1, 1]$ is a random variable, and $A$ is a positive semi-definite matrix. Is $E[X]E[AX] - E[XAX]$ positive semi-definite?

Let $X \in [-1, 1]$ denote a random variable. Let $A \in \mathbb{R}^{n \times n}$ denote a random positive semi-definite (PSD) matrix. $X$ and $A$ are dependent. Let $E$ denote expectation. My ...
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If $T^2$ is positive(that is $T^2$ is self-adjoint and $\langle T^2v,v\rangle\geq 0$) , when $T$ can be self-adjoint?

Suppose $V$ is a vector space with dimension $n$, I'm doubting that if operator $T^2$ is positive, then $T$ under some situations can be self-adjoint. However, I'm not certain about my reasoning. My ...
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Show that the difference between a PSD matrix and the block diagonal of its principal sub matrices is also PSD

Let $A \in \mathbb{R}^{n \times n}$ denote a positive semi-definite matrix (PSD). Let $A_{n_1} \in \mathbb{R}^{n_1 \times n_1}$ denote the principal submatrix of $A$ that corresponds to taking the ...
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Let $A$ be a PSD matrix, and $B$ a diagonal matrix where all diagonal entries are between 0 and 1. Is $A - BAB$ PSD?

Let $A \in \mathbb{R}^{n \times n}$ be a positive semi-definite (PSD) matrix, and $B \in \mathbb{R}^{n \times n}$ a diagonal matrix where all diagonal entries are between 0 and 1. Is it true that $A - ...
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If the operator $P$ is positive and $T$ is self-adjoint, then is there exists a positive number $n$ such that $nP+T$ is positive?

$V$ is a finite-dimensional complex inner product space and $P, T\in L(V)$. If the operator $P$ is positive and $T$ is self-adjoint, then is there exists a positive number $n$ such that $nP+T$ is ...
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When does $PAP = (Q \otimes R) A (Q \otimes R)$ imply $P$ is a tensor product, for orthogonal projections $P$, $Q$, and $R$?

Suppose $A$ is a positive semi-definite operator on $V \otimes W$ such that $\operatorname{tr}(A) > 0$, where $V$ and $W$ are finite-dimensional complex vector spaces. Let $P$ be an orthogonal ...
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Prove the nonzero postive operator $P$ has 1 dimensional range

Let $V$ be a finite-dimensional complex inner product space, and let $P \in L(V)$ be a non-zero positive operator with the property that for all positive operators $Q, R$ such that $P = Q+R$, there is ...
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Is there any hint how to prove this?

Let's consider matrix M is defined as follows: $M = \begin{bmatrix} P & v \\ v^t & d \end{bmatrix}$, where $P \succ 0$, d is a scalar, and v is a vector. Problem: :To $M \succ 0$ be a ...
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A question on inequalities associated to hermitian positive semidefinite matrices

Let $\mathcal{H}_N$ denote the cone of hermitian $N$ by $N$ positive semidefinite matrices and let $H \in \mathcal{H}_N$. We associate to $H$ the following complex numbers: \begin{align*} c_1 &= ...
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Is the matrix for Cholesky decomposition semidefinite or definite?

So we have a matrix $A$. We need to check if it is positive definite (numerically). At lectures we have done that with brute-force Cholesky decomposition, and if any of the square roots is not defined,...
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Relationship between PSD matrix rank and its convexity

I know that the matrix $W=vv^T$ is PSD and of rank 1. I also know that the set of rank-1 matrices is not convex. My question may be trivial but I would appreciate your feedback. If $W$ is PSD, it ...
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If $A$ is positive semidefinite and $P$ is unitary, is $B = P^{-1}AP$ positive semidefinite?

I know that in general, positive semidefiniteness is not preserved by matrix similarity. But is it preserved when $B = P^{-1}AP$ and $P$ is unitary?
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Is it possible that $A+A_{\text{off-diag}}\succ0$ , where $A\succneqq0$?

Denote $A_{\text{off-diag}}:=A-\text{diag}(A)$, i.e. setting diagonal elements to be zero. Denote $A\succneqq0$ iff $A$ is positive semidefinite (and not positive definite), while $A\succ0$ iff $A$ is ...
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Given any symmetric matrix $A$, how to find a diagonal matrix $B$ such that $B \succeq A$?

Given any symmetric (not necessarily PSD) matrix $A$, how can we find a good diagonal matrix $B$ such that $A \preceq B$? We want each of the diagonal elements of $B$ to be as small as possible. One ...
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Is this rank-$1$ (complex) matrix positive semidefinite?

In Is this rank-$1$ matrix semidefinite?, I have seen that $X = xx^T$ is PSD when $x$ is real. What about the case when $X$ is Hermitian? I know that it is PSD but I'm not exactly sure how to prove it....
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How to prove that changing the equality constraints does not affect the sign of the optimal value of the objective function?

Given random Hermitian matrices $A_1,A_2,A_3,A_4$,they satisfy: $$ \text{tr}(A_i)\in[0,1],\quad i=1,2,3,4.\quad \text{tr}(A_1+A_2)=\text{tr}(A_3+A_4)=1 $$ Given Hermitian variables $X_1,X_2,X_3,X_4$, ...
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Lower bound for $\| (I+A)^{-1} - (I+B)^{-1}\|$ where $A, B \ge 0$

Given two positive semi-definite matrices $A,B \ge 0$, I am interested in finding a lower bound for the operator norm: $$ \| (I+A)^{-1} - (I+B)^{-1}\|. $$ So far I have only been able to find an upper ...
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Schur complement and positive semidefinite cones

I know that my question might be trivial but I would appreciate your feedback. I know that the Schur complement can be used to express a quadratic inequality as a positive semidefinite matrix and vice-...
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Cholesky decomposition for symmetric positive semi-definite matrices

On page 5 here: https://stanford.edu/class/ee363/lectures/lmi-s-proc.pdf $A$ and $B$ are decomposed into $A^{1/2} A^{1/2}$ and same for $B$. Is this from Cholesky decomposition? Can someone prove ...
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When we have a universal approximator, can we approximate any nonnegative function by passing the approximator to ReLU activation?

Consider the domain of all functions that will be mentioned is $X$, a compact subspace of $\mathbb{R}^n$. Suppose that, given a continuous function $f$, $\forall \epsilon > 0$, $\exists \hat{f} \in ...
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Positivity and tensor product

A matrix $M$ acting on an inner product space $V = \mathbb{C}^n$ is called positive-semidefinite if, for every $v \in V$, $\langle Mv,v \rangle \ge 0$. Consider the inner product space $V \otimes V$ ...
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Positive semidefiniteness of matrix $\mathbf{G}$

Suppose $\mathbf{X}$ and $\mathbf{Y}$ are square, real, and symmetric matrices. $\mathbf{X}$ is positive definite and $\mathbf{Y}$ is positive semidefinite. $a$ and $b$ are positive scalars, and $\...
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Characterization of matrices with same column spaces

Given $G = \begin{pmatrix} A & B \\ B^T & C \end{pmatrix}$ symmetric positive semi-definite and $B$ symmetric positive semi-definite, is there a way to impose that \begin{equation} ...
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Is the positive operator $TT^*-T^*T$ on a finite complex inner product space is a zero operator?

Suppose the vector space is a complex finite-dimensional inner product space $V$, and the operator $TT^*-T^*T$ is a positive operator on it. Can this information be sufficient to conclude that $TT^*-T^...
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how to prove that eigenvalues of a matrix is in unit disk

Define block matrix with real entries $$A=\begin{bmatrix}I-\alpha H-\alpha\beta L &-\beta I \\ \alpha L & I \end{bmatrix}$$ where $H,L$ are real symmetric positive semi-definite (may be ...
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How to extract the vector from a rank-$1$ matrix? [closed]

Given Hermitian and positive semidefinite rank-$1$ matrix ${\bf Z} \in \mathbb C^{N \times N}$, how to find vector $\mathbf z \in \mathbb C^{N \times 1}$ such that $\mathbf Z = \mathbf z \mathbf z^{H}$...
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Cholesky decomposition for a Hermitian matrix in SDP

I have a variable matrix $W$ that is Hermitian and is used in two SDP problems. Problem 1 has constraints that depend on the real diagonal elements of $W$. Example of the constraint is $W_{ii}+x_{ij}...
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Is it true that for all trace preserving positive maps $\phi$ that $\phi^\dagger (\rho)\leq 1$, where $Tr(\rho)=1$ and $\rho$ is semi-positive?

I would conclude it from the following reasioning: Take $\rho$ and $Y$, both semi-positive. We have (with Cauchy-Schwarz inequality) $Tr(Y \phi^\dagger(\rho))= Tr(\phi(Y)\rho)\leq Tr(\phi(Y))Tr(\rho)=...
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Determine whether the matrix is positive/negative semidefinite/definite or indefinite.

Determine whether the matrix is positive/negative semidefinite/definite or indefinite: $$\left(\begin{array}{cccc} 2&2&0&0\\ 2&2&0&0\\ 0&0&3&1\\ 0&0&1&3 ...
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Positive semidefinite matrix with 0 on diagonal

If I have a Hessian matrix $ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $ might someone help me understand why this is not positive semidefinite? My understanding that if for ...
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On Sylvester’s criterion for $2 \times 2$ matrices

In my electrical engineering text, I have an expression of the form: $$E_m(x_1, x_2) = \frac{1}{2}(A_{11}x_{1}^2+2Mx_1x_2+A_{22}x_{2}^2) > 0$$ where $$A = \begin{bmatrix} A_{11} & M \\ M &...
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