Questions tagged [matrix-analysis]

The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.

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18 views

Approximating singular values of the resolvent matrix for a non-Hermitian matrix

I have a pretty niche question that stems from the following answer: https://mathoverflow.net/a/79129/87974 I am interested in bounding the following quantity: $b := |e_k^T R(z)^T R(z) e_k|$, where $...
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Characterizing a variant of completely-Q matrices in linear complementarity problems

Given an $(n\times n)$-matrix $A$ (over the reals) and a vector $q \in R^n$, the linear complementarity problem $LCP(A,q)$ is the following problem: Find $w,z \in R^n_+$ such that $$w = q + Az,$$ and $...
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1answer
89 views

The rank of the Hadamard product

For matrices $D\in C^{d×p}$ and $E\in C^{d×p}$ with $d>p$, if $D$ is a full column matrix, for what condition that $D \odot E$ is also a full column matrix where $\odot$ denotes the Hadamard ...
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1answer
76 views

Operator norm of difference of matrix decompositions

This question is in part related to a question that I have already posed. Say I have two symmetric positive definite matrices and their respective Cholesky decompositions $\mathbf{A} = \mathbf{L}_A \...
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17 views

Meaning of $K'[\cdot]$ when $K$ is an symmetric Onsager (matrix) operator

My question is from the section 5.2 of the monograph "Entropy Methods for Diffusive Partial Differential Equations" written by Ansgar J$\ddot{\text{u}}$ngel. To be specific, I do not know ...
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60 views

Uniform continuity of Cholesky decomposition

I am trying to figure out whether the Cholesky decomposition is uniformly continuous within a set of matrices with bounded entries. Specifically, I would like to derive a modulus of continuity for the ...
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26 views

Projecting matrix from LDL^T factorization

When factorizing a real symmetric matrix $A$ into $LDL^T$, the matrix $D$ can have 1x1 or 2x2 blocks on the diagonal. A condition for $A$ to be positive-definite is that all 1x1 and 2x2 blocks of $D$ ...
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1answer
186 views

Kulkarni-Nomizu square root of the Riemann tensor

Given a Riemann tensor $Riem$, what are conditions such that $Riem=B\star B$ for some bilinear symmetric form $B$, where $\star$ is the Kulkarni-Nomizu product? It follows from the proof of ...
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141 views

System of matrix equations

Problem definition: Let $x_i \in \mathbb{R}^d$ and $a_i \in [0,1]$, for all $i = 1,\dots, k$ (with $k\geq d$). Define $M(a) = \sum_{i = 1}^k a_i x_ix_i^T,$ and assume $M(a) \succ 0.$ Question: Is ...
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144 views

Derivative of log determinant [closed]

Let $x_i \in\mathbb{R}^d$ and $a_i\in [0,1]$ for $i = 1,\dots,k$. How to compute the following derivative? $$ \frac{d}{da_j}\log \det\left(\sum_{i = 1}^k a_ix_ix_i^\top\right). $$
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72 views

Is there a specific name for this optimization problem?

Let $A$ be an $n\times n$ symmetric positive definite matrix with eigenvalues and eigenvectors $\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_n>0$ and $v_1,v_2,\cdots,v_n$ respectively. We know that the ...
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1answer
45 views

Upper bound on the size of vectors contained in an ellipsoid?

Crossposted at Math SE Consider the diagonal matrix $$D=\left[\begin{array}{cccc} 1^{- 2 p} & 0 & \cdots & 0 \\ 0 & 2^{-2 p} & \cdots & 0 \\ \vdots &...
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1answer
92 views

Matrix equation involving quadratic form

Let $X,Y\in\mathbb{R}^{n\times k}$, $\Lambda(\alpha) = \text{diag}(\alpha)$, with $\alpha\in\mathbb{R}^k$, and let $c,d\in\mathbb{R}^+$ be positive constants. Let $$A_i(\alpha) = (X\Lambda(\alpha) X^...
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34 views

Growth of the number of columns $j=1,\dotsc,p$ such that $\|Ae_j\|_1 > p^\alpha$ for symmetric $A$ with bounded spectrum?

Consider the set $\mathcal S(p)$ of symmetric matrices $A$ of size $p\times p$ with bounded spectrum, say, $\lVert A\rVert_\text{op}\le 10$ and $\lVert A^{-1}\rVert_\text{op}\le 10$. Let $\alpha>0$ ...
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91 views

Multiplication of two Pauli string

Given a Pauli string $P_i \in \{ I,X,Y,Z\}^{\otimes n} $ Example: $P_0 = XXYIZ = X \otimes X \otimes Y \otimes I \otimes Z $. Here $I,X,Y,Z$ are Pauli matrices defined explicitly as: $$ I = \begin{...
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Derivative of Cholesky Factor not invertible

This question refers to the following previous post The derivative of the Cholesky Factor. It is said there that \begin{equation} \frac{\mathrm{d} \operatorname{vec}\left(XX^{\top}\right)}{\mathrm{d} \...
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Distance of low-rank matrices to the identity for the $\infty$-norm

I am trying to get a lower bound (or even the exact value) of $$ \min_{X \in \mathbb{R}^{n\times n}} \|X - I_n\|_{\infty} \enspace \text{s.t.} \enspace \mbox{Rank}(X) = m $$ where $m \leq n$, and the ...
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Stationary distribution of a weighted directed acyclic graph

Is there any way to calculate the equilibrium (stationary) distribution for a weighted directed acyclic graph? Some references emphasized adjacency matrix to be symmetric. https://arxiv.org/abs/1012....
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57 views

Solving a fully nonlinear first order PDE

given a symmetric matrix of Holder continuous functions $A(x)$ such that $$ \frac{1}{C} |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq C |\xi|^2 $$ find a vector field $\Phi$ such that $$ D \Phi(x)^t D ...
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204 views

Gradient Descent for Markov Dynamics [closed]

The closed-loop dynamics of a linear optimal controller are simple but have interesting properties. From a starting state $\mathbf{v}(0)$ the dynamic can be iterated to reach a final state $\mathbf{v}(...
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50 views

Square root of a circulant matrix block

I'm trying to show the following: Given the following $n\times n$ symmetric circulant matrices $$A^*=\begin{pmatrix} 1 & -\mu_a & 0 & ...&0&-\mu_a \\ -\mu_a & 1 & -\mu_a &...
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References for matrix variational problems on the unitary group

Let $U(d)$ denote the group of unitary $d\times d$ matrices. Let $\mathcal C_d$ denote the cone of Hermitian positive semidefinite $d\times d$ matrices. Fix an integer $r\geq 1$ and let $C(U(d),\...
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49 views

State-dependent positive definite matrix

Consider a function $f(\mathbf{x})=\mathbf{M}_\mathbf{x}$ that outputs a nonsymmetric matrix $\mathbf{M}_\mathbf{x} \in \mathbb{R}^{N \times N}$ given an input vector $\mathbf{x} \in \mathbb{R}^N$. Is ...
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Is a basis that almost diagonalizes a matrix 'close' to its eigenbasis?

Let $A\in\mathbb{R}^{n\times n}$ be diagonalisable (over $\mathbb{C}$) with pairwise distinct eigenvalues $\lambda_1,\ldots,\lambda_n\in\mathbb{C}$, and suppose that $$\tag{1}S^{-1}\cdot A\cdot S = \...
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21 views

Proof negative-definiteness of a nonsymmetric and rank-deficient matrix

Consider the vectors $\mathbf{a} \in \mathbb{R}^N$ and $\mathbf{b} \in \mathbb{R}^N$ with $N>1$ and $\mathbf{a} \neq \mathbf{b}$. The product $\mathbf{C}=\mathbf{a} \mathbf{b}^T \in \mathbb{R}^{N \...
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64 views

Infinite positive matrices with probability eigenvector

Let $A$ be an infinite non-negative matrix with integer entries ($a_{ij} \geq 0, \forall i,j \in \mathbb N$). Suppose that $A$ is irreducible, aperiodic, and recurrent. So that it satisfies the ...
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1answer
71 views

How do you solve this quadratic matrix equation?

could you please help me solve this quadratic matrix equation? I look around, seems like there is no general solution for it.. $$-BX^2 + X - C = 0$$ for X, B and C are (3x3) matrices. B and C are ...
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154 views

The d(abc)-theorem, the abc-conjecture and positive definite kernels over the natural numbers?

I noticed in the work of Hector Pasten, Th.1.11 the $d(abc)$ theorem and have a question, after doing some experiments with sagemath. Let $s_k(n) = \sum_{d|n}{ d^k }$ be the sum of divisiors $d$ of $n$...
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80 views

How to design a matrix function to meet the following conditions

Suppose I have a matrix $X\in \mathbb{R}^{n\times n}$, such that $X$ is symmetric Do not know the rank I want design a matrix function $f(X,Q)\in \mathbb{R}^{n\times n}$ with $Q = qq^T$ and $q\in \...
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124 views

If $x \ge 0$ and $\mathbf{1}^Tx \le \|x\|^2$ then $\mathbf{1}^T(I - xx^T / \|x\|^2) \mathbf{1} \ge \| [\mathbf{1} - x]_+ \|^2$

Notation. Denote $\mathbf{1}=(1,1,\ldots,1)$ as the vector-of-ones in $\mathbb{R}^n$. Write the "positive part" as $[\alpha]_+ = \max\{\alpha,0\}$ for $\alpha\in\mathbb{R}$ and $[(x_1,x_2,\...
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1answer
93 views

Uniform smoothness inequality for Schatten norms

I've previously asked this question on stack exchange. I'm looking for a proof of the inequality $$ \left[ \frac12(\left\|A+B\right\|_p^p + \left\|A-B\right\|_p^p)\right]^{2/p} \leq \left\|A\right\|_p^...
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Parseval's equivalent of Norm that includes a Projection matrix

I need to optimize the norm, ${\bf x}^H {\bf P}_{\bf B} {\bf x} $, where, ${\bf P}_{\bf B} = {\bf B}^H({\bf B} {\bf B}^H)^{-1} {\bf B}$, ${\bf B}$ is a known $M \times N$ matrix, with $M < N$ and $...
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1answer
171 views

Growth of eigenvalues for certain sequences of matrices

Suppose we have an aperiodic matrix $A_t$ that has entries that are either $0$ or are positive integer powers of $t$, i.e. we could have $$A_t = \begin{pmatrix} 0 & t & t^2\\ t & t^2 &...
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1answer
598 views

Counting eigenvalues without diagonalizing a matrix

Today's arXiv has a paper by Pierpaolo Vivo, Index of a matrix, complex logarithms, and multidimensional Fresnel integrals, which asks the question whether it is possible to calculate the number $N(\...
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Is this lower bound on the singular values of the sum of two matrices correct?

Equation 7 of this paper (Ramazan Türkmen, Zübeyde Ulukök, Inequalities for Singular Values of Positive Semidefinite Block Matrices, International Mathematical Forum, Vol. 6, 2011, no. 31, 1535 - 1545)...
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A lower-bound on matrix-function with vector product

I am currently trying to show that a sequence of homeomorphisms converges to some limiting homeomorphism using Anderson's the inductive convergence criterion. However I can't explicitly compute the ...
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Expectation of random matrix (minimum positive eigenvalue)

Assume, $M_i$ are random symmetric positive semi-definite matrices and there exists $ 0 < \mu_1 \leq \mu_2 $ such that $\mu_1 \|x\|^2 \leq x^TE[M_i]x \leq \mu_2 \|x\|^2$ holds for some $0 \neq x \...
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175 views

Follow up: Show that these vectors are linearly independent almost surely

I posted this question some time ago here. I started a bounty for it and received an answer which helped me a lot. However, I still have some issues I wan't to discuss regarding it. Unfortunately I ...
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1answer
131 views

Characterize matrix range

$\DeclareMathOperator{\col}{\operatorname{col}}\DeclareMathOperator{\diag}{\operatorname{diag}}\DeclareMathOperator{\Range}{\operatorname{Range}}$Let $A \in \mathbb{R}^{n\times m}$, $D = \diag(d) = \...
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1answer
582 views

Show that these vectors are linearly independent almost surely

So I'm doing research in control theory and I have been stuck with this problem for a while. Let me explain my issue, then my proposal, and finally my concrete question. Problem: I have $m<n$ real $...
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117 views

L_q matrix inequality

The following arose out of studying $\ell_q$ Lewis weights. Let $P$ be a real $n \times n$ orthogonal projection matrix (i.e., $P$ is symmetric and $P^2 = P$) and let $W$ be the diagonal matrix ...
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Can anything be said about $X^T C X$ based on the structure of $C$?

I want to analyze the following expression ($\mathbf{X} \in \mathbb{R}^{N\times P}, N \gt P$): $$ \operatorname{Trace} (\mathbf{X}^T \mathbf{C} \mathbf{X})^{-1} $$ where $\mathbf{C}$ is a block-...
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158 views

$A\geq B\Rightarrow A^{-1}\leq B^{-1}$ entrywise for pos.def. symmetric matrices?

My question follows from https://math.stackexchange.com/questions/3857976/inverse-inequality-of-symmetric-matrix. Suppose we assume that $A$ and $B$ are two positive definite matrices with positive ...
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1answer
173 views

How to find upper and lower bound

Let $\Sigma \in S_{++}^n$ be a symmetric positive definite matrix with all diagonal entries equal to one. Let $U \in \mathbb{R}^{n \times k_1}$, $W \in \mathbb{R}^{n \times k_2}$, $\Lambda \in \mathbb{...
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0answers
93 views

Bounding Frobenius norm of pseudo-inverse

$\DeclareMathOperator{\F}{\mathrm{F}}$Let $\mathbf{A}$ and $\mathbf{A}^\prime$ be two $m\times n$ matrix such that $\|\mathbf{A}-\mathbf{A}^\prime\|_{\F}\leq \delta$. Is there any bound for the ...
6
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1answer
340 views

Condition number for a symmetric positive definite matrix

I am trying to get an estimate for the induced 2-norm condition number $\kappa_{2}(M)$ of this matrix $M$: $$M_{ij} = \frac{1}{(n-i)!(n-j)!(2n-i-j+1)} = \displaystyle\int_{0}^{1}\frac{x^{n-i}}{(n-i)!} ...
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57 views

Identify an ordered-eigenvalue simplex with tetrahedral dihedral angle $\cos ^{-1}\left(\frac{1}{3}\right)$ in volume, area formulas

Let us order the four eigenvalues ($\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4$, $\Sigma_{i-1}^4 \lambda_i=1$) of a Hermitian, trace-one, positive-definite $4 \times 4$ matrix $\rho$ (a “...
2
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1answer
137 views

Necessary conditions for existence of linear combination of these matrices to be singular

I'm doing some research in Control theory, and a stumbled with this problem. Any help is appreciated. QUESTION Let $P_1,\dots,P_m$ be $m$ symmetric positive definite $n\times n$ matrices with $m<n$ ...
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50 views

Approximation bounds for matrix multiplication

$\DeclareMathOperator{\op}{\mathrm{op}}$Since matrix multiplication is continuous, I expect that if $A_n\to A\in \mathrm{Mat}_{d\times d}(\mathbb{R})$ for the operator-norm and if $x_n\to x$ in $\...
8
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2answers
239 views

When does $\left\Vert f(\mathbf{N}) - f(\mathbf{M})\right\Vert_{\mathrm{op}} \leq k\left\Vert \mathbf{N} - \mathbf{M}\right\Vert_{\mathrm{op}}$ hold?

Define the Frobenius norm of a matrix as $\left\Vert A \right\Vert_{\mathrm{F}}=\sqrt{\sum_{i,j} A_{ij}^2}$ and the operator norm as $\left\Vert A \right\Vert_{\mathrm{op}}=\sup_{x \not = 0} \frac{\...

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