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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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Proof that $\ln \det(\mathbf I + \mathbf X\mathbf Y^{-1}) = \ln \det (\mathbf I + \mathbf Y^{-1/2}\mathbf X \mathbf Y^{-1/2})$

How do we prove that \begin{equation} \ln \det(\mathbf I + \mathbf X\mathbf Y^{-1}) = \ln \det(\mathbf I + \mathbf Y^{-1/2}\mathbf X \mathbf Y^{-1/2}). \end{equation} Here $\mathbf I$ is the identity ...
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Is it true that $tr(APA^T)$ > $tr(AQA^T)$, if $tr(P)$ > $tr(Q)$

Assuming $P$ and $Q$ and positive definite matrix. Is it true that $tr(APA^T)$ > $tr(AQA^T)$, when $tr(P)$ > $tr(Q)$. [EDIT after first answer: not just that, actually $P_{ii} > Q_{ii} $ for ...
zvi's user avatar
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singular positive semi-definite matrix in electromagnetism

anyone knows where he drew this conclusion from?
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Is the Wigner-Ville representation of a non-negative operator a non-negative time-frequency density?

Consider a non-negative operator $K \geq 0$ acting on signals $x$ (say a complex Hilbert space). The associated quadratic form can take the following temporal representation: $$\langle x , K x\rangle =...
Alexandre's user avatar
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Off -diagonal perturbations of a positive semi-difinite matrix

Does the following statement hold? For a positive semi-definite(PSD) matrix X, if $rank(X) \ge 2 $, there exists a nonzero off-diagonal matrix D that both X+D, X-D are also PSD. If it doesn't work, ...
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Does isometry on PSD matrices preserve eigenvalues?

Let $S$ be the set of symmetric matrices and $T: S\rightarrow S$ be a linear isometry. Moreover, let $T$ be a bijection from the space of PSD matrices to the set of PSD matrices. Must $T$ preserve ...
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Reference for uniqueness of vector realizations of a Gram matrix

Let $v_i,\ldots,v_n\in\mathbb R^k$, and let $G_{ij} = \langle v_i, v_j\rangle$ be their Gram matrix. As Wikipedia notes, the Gram matrix $G$ uniquely specified the geometry of the $v_i$: if there is a ...
E.P.'s user avatar
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Positive-semidefiniteness of complex Teoplitz Hermitian matrix and implications on its range space

I have a problem understanding one proof in this article (Theorem 6.3 p.31). Let $x>0, \mathbf{z}\in\mathbb{C}^n$, $\mathbf{u}=\begin{bmatrix}u_1\\u_2\\\vdots\\u_n\end{bmatrix}\in\mathbb{C}^n$, and ...
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Truncate off-diagonal values of PSD matrices results in PSD?

Suppose I have a PSD matrix $X$ with 1 on the diagonal and all off-diagonal values with magnitudes less than or equal to 1. Suppose I truncate some of the off-diagonal values by some small $\epsilon$ (...
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Inequality involving matrix trace and diagonalisable matrices

Given two real PSD matrices diagonalisable by orthogonal matrices: $A=UDU^T$ and $B=VEV^T$, prove that $$tr(A+B-2(A^{1/2}BA^{1/2})^{1/2})\geq0.$$ We can rewrite the inequality as $$tr(D)+tr(E)\geq tr((...
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$A-B$ semi positive definite can tell us the information of the positive inertia index

Suppose $f,g$ are symmetric real quadratic forms on $\mathbb{R}^n$ such that $f(x)\geq g(x)$ for all $x\in\mathbb{R}^n$. Prove that the positive inertia index of $f\geq$the positive inertia index of $...
Louis Wiles Young's user avatar
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Smallest eigenvalue of $A^T D^{-1} A + D$ for positive diagonal $D$.

I am wondering whether or not there exists a way to analyze the eigenvalues of $$A^T D^{-1} A + D$$ for a square matrix $A \in \mathbb{R}^{n \times n}$ and positive diagonal matrix $D\in \mathbb{R}^{n ...
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Positive semidefiniteness of Laplacian of undirected graph

Let $D$ be a directed graph. Given a directed edge $e:x\to y$ of $D$, the head of $e$ is defined to be the vertex $y$ and its tail is defined to be the vertex $x$. The gradient matrix of $D$, denoted $...
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Proving positive semidefiniteness of a matrix multiplication

Assume $Q \in \mathbb R^{n\times n}$ is positive semidefinite and $I$ is the identity matrix. How can I prove that the matrix $$M = (I + \alpha Q)^{-1} Q $$ is positive semidefinite for all $\alpha \...
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Limit of $\mathrm{tr}(P(\alpha QP + I)^{-1})$ as $\alpha \to \infty$?

Say that $P, Q$ are two real, symmetric positive semidefinite, possibly singular matrices. What is the limit $$ \lim_{\alpha \to \infty} \mathrm{tr}(P(\alpha QP + I)^{-1})? $$ Simultaneously ...
Drew Brady's user avatar
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Maximal value of $\sum_{n=1}^N z^T X(P_n X + I)^{-1}z$ over positive semidefinite cone?

Let $z$ denote a unit vector. Fix a finite collection of positive semidefinite matrices $\mathcal{P}$. Define the function and set $$ f_{\mathcal{P}}(X) = \sum_{P \in \mathcal{P}} z^T X(PX + I)^{-1} z,...
Drew Brady's user avatar
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Positive semi-definiteness of specified covariance matrix

I am working with model where the covariance matrix will be of the form $$\boldsymbol\Sigma=\begin{pmatrix} \omega_1^2\boldsymbol C & \omega_1\omega_2\boldsymbol C\\ \omega_1\omega_2\boldsymbol C &...
Remy's user avatar
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Is a $2\times 2$ block matrix with PSD matrices on the off-diagonal and diagonal matrices on diagonal PSD?

I am working with a model where the covariance matrix is of the form $$\boldsymbol\Sigma=\begin{pmatrix}A&B^\top\\ B&C\end{pmatrix},$$ where $A=\sigma^2\mathcal I$, $C=\tau^2\mathcal I$, and $...
Remy's user avatar
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Why is the function $x\mapsto |x|^a$ ($a\in ]0,2]$) negative definite?

I'm studying negative definite functions from the book C. Berg, and G. Forst, Potential Theory on Locally Compact Abelian Groups (Springer Berlin Heidelberg, Berlin, Heidelberg, 1975). In this book, a ...
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$F(x_1,\ldots, x_n) = \sum_{i=1}^m \big( \sum_{j=1}^n c_{ij} x_j\big)^2 \overset{?}{\Leftrightarrow} F$ is a positively semi-definite.

Main question. $F(x_1,\ldots, x_n) = \sum_{i=1}^m \big( \sum_{j=1}^n c_{ij} x_j\big)^2 \overset{?}{\Leftrightarrow} F$ is a positively semi-definite form of $x_1,\ldots,x_n$. It's possible that $c_{ij}...
Sergei Nikolaev's user avatar
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Why the first matrix is not always psd but the second matrix is always psd?

I learned that one important motivation for the Newey-West covariance estimator is that naive estimator of the covariance matrix is not necessarily positive semidefinite(psd from now on) (see the ...
ExcitedSnail's user avatar
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Trace of product of positive semi definite matrix and matrix with all negative eigenvalues

Suppose $A \in \mathbb{R}^{n \times n}$ is a Hurwitz matrix, that is, all the eigenvalues of $A$ have strictly negative real parts, and that $S \in \mathbb{R}^{n \times n}$ is a symmetric positive ...
sixtyTonneAngel's user avatar
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Example of semi algebraic subset over a real closed field

I am trying to find interesting example of semi algebraic subset over a real closed field other than those we have by considering $\mathbb{R}^n$. I tried to construct an example involving matrices. ...
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Trace of symmetric part of product of 3 positive semidefinite matrices

Suppose we have 3 symmetric positive semi-definite matrices $A,B,C$. How can one prove or give counter-example for the following statement? $$ \mathrm{trace}(AB(A-C)^2 + (A-C)^2BA) \ge 0 $$ where we ...
sixtyTonneAngel's user avatar
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A case problem about rank-1-perturbation of diagonal matrices

I have the following prediction for rank-1 perturbations of diagonal matrices, but I don't know how to prove (or disprove it). Given $v:= [v_1,...,v_K] \in (0,1]^K$, we define $a:= \sum_{k=1}^K v_k = \...
abcxyzf's user avatar
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Lower bound of the spectral gap of positive semi-definite matrices

Let $M$ be an $n \times n$ positive semi-definite, symmetric real matrix such that $|| M ||_2 = 1$ (with $||\cdot ||_2$ being the Schatten 2-norm) and $\dim \ker M = k < n$. Let the spectrum of $M$,...
incud's user avatar
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Power series of matrices that is similar to hyperbolic sine

Consider a power series $$ \phi = \sum_{k=0}^{\infty} \frac{(M_2M_1)^kM_2}{(2k+1)!} $$ where $M_1$ and $M_2$ are symmetric positive semidefinite matrices, thus standard square roots $M_1^{\frac{1}{2}}$...
Eric J's user avatar
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2 answers
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Uniqueness of Hessenberg matrix in Cholesky factorization of Hankel matrix

I'm reading the paper how bad are Hankel matrices?, where the author claimed the following: Lemma 2.1. For any positive definite Hankel matrix $H \in \mathbb{R}^{n \times n}$ there exist a ...
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When is $\Vert A+B \Vert = \Vert X + X^\dagger \Vert$ for $[A, X; X^\dagger, B] \succcurlyeq 0$?

Given $$ H = \begin{pmatrix} A & X \\ X^\dagger & B \end{pmatrix} \succcurlyeq 0 , $$ with $A \succcurlyeq 0$ meaning $A$ is positive semi-definite (and Hermitian) and $A, B, X$ are $n\...
Aritra Das's user avatar
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Limit of Eigenvalues of Matrix Expression

I would like to study the limit $$ \lim_{x\rightarrow \infty} \lambda_{\min} [x(I_n + xA)^{-1}] , $$ where $x \in \mathbb R_{\geq 0}$ is a scalar, $I_n$ is the $n\times n$ identity matrix and $A \in \...
Trb2's user avatar
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1 answer
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Order of positive matrices on a subspace and inverse

Let $A$ and $B$ be two positive semidefinite symmetric matrices such that for all $x$ in the range of $B$, $x^{\top}Ax \leq x^{\top}Bx$. I am wondering if the following holds true: for all $\lambda &...
DimSum's user avatar
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Operator norm of the sum of two positive semidefinite matrices

Consider two positive semidefinite and symmetric matrices, namely $\bf{A}$ and $\bf{B}$. Denote $\Vert\cdot\Vert$ as the operator norm. If we have another positive semidefinite matrix $\bf{B}^*$ that ...
TNLI's user avatar
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2 votes
1 answer
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What is the reason for endowing the positive semidefinite cone with the Affine Invariant Riemannian Metric instead of a Euclidean metric?

I came across this post Distances defined in manifold of symmetric positive definite matrices because I had the same questions. I did not understand the answers provided, so I wanted to try to answer ...
user19402204's user avatar
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subset of positive matrices such that the Loewner order is total

Are there known characterizations of subsets of positive matrices over which the Loewner order is total ?
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A sort of extension of Cauchy-Schwarz inequality?

Let $a_{i}$ and $b_{i}$, $i = 1,\dotsb,n$ be real numbers. Denote $S_{pq}:=\sum_{i=1}^{n}a_{i}^{p}b_{i}^{q}$. I want to show that, $$ (S_{20}S_{02} - S_{11}^2)S_{00} - (S_{10}S_{02} - S_{01}S_{11})S_{...
Don Lee's user avatar
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Is $a b^T$ + $b a^T$ positive semidefinite if $a^T b > 0$ [closed]

Let $a$ and $b$ are $m \times 1$ constant vectors such that $a^T b > 0$. I am wondering if the matrix $a b^T$+$b a^T$ is positive semidefinite. Thanks
Charles Chou's user avatar
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Sum of positive semi-definite matrix and positive definite matrix?

Is the sum of a positive semidefinite matrix and positive definite matrix a positive definite matrix? I have a positive semidefinite matrix $M\in \mathbb{R}^{n\times n}$ and the identity $I_n$, is ...
atul ganju's user avatar
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1 answer
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why this equality involving matrix holds true?

I am studying Lemma 11 of the paper I am having difficulty understanding on the last step, in particular, I have two questions: The first question (1) On the first equality of the last step on page 15,...
chloe's user avatar
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how to show that a nonsingular positive semidefinite matrix is positive definite?

Got a question regarding the following statement: a nonsingular positive semidefinite matrix is positive definite. Is this true? If yes, how to show it. If the above statement is not true, how about ...
ExcitedSnail's user avatar
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Prove $\ker(A+T(A))\subseteq \ker(A)$ for $A\geq 0$ and $T$ positive linear map

Let $A\geq 0$ be a positive semi-definite complex matrix in $M_d(\mathbb{C})$. Let $T:M_d(\mathbb{C})\to M_d(\mathbb{C})$ be a positive linear map between $d\times d$ complex matrices, i.e., $A\geq 0\...
Evangeline A. K. McDowell's user avatar
3 votes
1 answer
32 views

Is this combination of positive semidefinite matrices positive semidefinite?

Say I have $A \succcurlyeq 0$, $ \begin{bmatrix}A & \vec{b}_1 \\\ \vec{b}_1^T & t_1 \end{bmatrix} \succcurlyeq 0 $, and $ \begin{bmatrix}A & \vec{b}_2 \\\ \vec{b}_2^T & t_2 \end{...
Pavel Komarov's user avatar
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Quadratic form with positive semi-definite matrix [closed]

Suppose $A$ is a positive semi-definite matrix, and $x$ and $y$ are real vectors. Under what conditions does the following result hold? $$ x'y > (<) 0 \Longrightarrow x'Ay \geq (\leq) 0 $$ ...
HXW's user avatar
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Decomposition of a positive semidefinite matrix in the form $Q = BB^H = CC^H$

Let $Q \in \mathbb{C}^{m \times m}$ be a given positive (semi)-definite matrix such that $Q = CC^H,$ for some known matrix $C \in \mathbb{C}^{m \times r}$ having full column rank and orthogonal ...
ketan bapat's user avatar
1 vote
0 answers
34 views

Random bounded triangular $T_n$ s.t. $Var[T_nS_nT_n^\top]\to 0$ for nonrandom psd $S_n$. Does $(T_n-\bar T_n)S_n\to^P0$ for some nonradom $\bar T_n$?

Let $T_n$ be a sequence of square random matrices with $T_n$ lower triangular with $diag(T_n)=(1,1...,1)$ and $S_n$ a sequence of deterministic symmetric psd matrices. All matrices are in $R^{d\times ...
jlewk's user avatar
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Prove convexity formula applied to positive semi definite matrices

I have to prove that : Let $p \in [0,1], A,B > 0$ and $\alpha \in [0,1]$. Then : $$(\alpha A+(1−\alpha)B)^p \ge \alpha A^p +(1−\alpha )B^p$$ I have the hint that : $X \ge Y \iff X−Y \ge 0$. I also ...
Ethan07's user avatar
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Inner product involving three PSD matrices

Let $\mathbb{S}^n_+$ denote the space of $n \times n$ symmetric positive semidefinite matrices. Let $A \in \mathbb{S}^n_+$ and $B \in \mathbb{S}^n_+$. Then $\langle A, B\rangle_F = \text{Tr}(A^\top B) ...
Student's user avatar
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1 answer
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Positive semidefinite inequality: $(AXA^T)^+ \geq (AXA^T + Y)^{-1}$ on $\textrm{Im}(A)$

I asked a question earlier but it wasn't quite correctly stated, so I'll reset. Let $A$ be an $n\times n$ matrix of rank $k<n$ and let $X,Y$ be two symmetric positive definite matrices. Let $Z^+$ ...
Moya's user avatar
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1 vote
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198 views

Prove that modified RBF function satisfies Mercer conditions.

Suppose that I have a modified RBF kernel function. $k(\mathbf{x},\mathbf{y}) = \exp{(-||\mathbf{x}-P\mathbf{y}||^2 })$ where $\mathbf{x},\mathbf{y}$ represent $d$ dimensional inputs and $P$ is the ...
flammmes's user avatar
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find the minimal positive eigenvalue satisfying (A+tB)v=0

Given conditions: matrix $A$ is symmetric and positive definite matrix $B$ is symmetric and NOT positive definite $0\leq t$ then linear system $$(A+tB)x=y$$ has solution for $0\leq t<\bar t$, ...
whitegreen's user avatar
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Limit of sum of factions of finite constants and matrix

Suppose $\mathbf{a}_{i}$ is a $2\times 1$ constant vector with $\frac{1}{N}% \sum_{i=1}^{N}\mathbf{a}_{i}\mathbf{a}_{i}^{T}$ converges to a finite $% 2\times 2$ matrix as $N\rightarrow \infty $, and $...
Charles Chou's user avatar

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