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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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 ...
Heinrich A's user avatar
1 vote
0 answers
62 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 $D$ ...
Toool's user avatar
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9 votes
1 answer
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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 ...
Carlo Mantegazza's user avatar
2 votes
0 answers
172 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 ...
Apprentice's user avatar
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1 answer
990 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? $$ \frac{d}{da_j}\log \det\left(\sum_{i = 1}^k a_ix_ix_i^\top\right). $$
Apprentice's user avatar
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110 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 ...
bernard's user avatar
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1 vote
1 answer
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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 &...
Peter's user avatar
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1 vote
1 answer
192 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^...
Apprentice's user avatar
1 vote
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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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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{...
KAJ226's user avatar
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5 votes
2 answers
208 views

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 ...
PAb's user avatar
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1 answer
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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....
Mehdi Nmz's user avatar
-1 votes
1 answer
109 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 ...
Harish's user avatar
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1 answer
222 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}(...
spencer wilson's user avatar
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107 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 &...
Giovanni Febbraro's user avatar
0 votes
1 answer
71 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 ...
Beram's user avatar
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0 answers
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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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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 \...
Beram's user avatar
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2 votes
0 answers
194 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 ...
SIB's user avatar
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1 answer
357 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 ...
Hung's user avatar
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1 answer
253 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 $n$...
mathoverflowUser's user avatar
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1 answer
104 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 \...
Denny's user avatar
  • 101
1 vote
2 answers
142 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,\...
Richard Zhang's user avatar
4 votes
1 answer
181 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^...
Florian Ente's user avatar
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123 views

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 $...
ashaw2's user avatar
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3 votes
1 answer
192 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 &...
Zestylemonzi's user avatar
8 votes
1 answer
736 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(\...
Carlo Beenakker's user avatar
2 votes
1 answer
513 views

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)...
Gabriele Oliva's user avatar
1 vote
0 answers
119 views

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 ...
ABIM's user avatar
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0 votes
1 answer
465 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 want to discuss regarding it. Unfortunately I can'...
FeedbackLooper's user avatar
2 votes
1 answer
155 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) = \...
Apprentice's user avatar
11 votes
1 answer
955 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 $...
FeedbackLooper's user avatar
0 votes
0 answers
146 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 ...
yangpliu's user avatar
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-1 votes
1 answer
172 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 ...
user812951's user avatar
2 votes
1 answer
414 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{...
newbie's user avatar
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2 votes
0 answers
558 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 ...
Math_Y's user avatar
  • 311
6 votes
1 answer
1k 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)!} ...
Abhishek Halder's user avatar
2 votes
1 answer
221 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 $m<n$ ...
FeedbackLooper's user avatar
1 vote
0 answers
68 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 $\...
ABIM's user avatar
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9 votes
2 answers
477 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{\...
Henry's user avatar
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1 vote
1 answer
197 views

How could I extend this result to a case where the matrices were not of full rank?

I'm reading this paper by Bhatia, Jain and Lim and on page 6 theorem 2, they state $$ \sqrt{\operatorname{tr}(A^{1/2}BA^{1/2})} = \max_{X>0} \{ |\operatorname{tr} X|: A\geq XB^{-1}X^{*} \} $$ where ...
Kashif's user avatar
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0 votes
0 answers
49 views

Non-square multiplication operator matrix

Let $A(x), x\in (0,1)$ an $2\times n$ matrix, with $n\geq 2$. Consider the multiplication operator $K$ on $[L^2(0,1)]^n$ defined as $$K: f(x) \mapsto Kf(x)=A(x).f(x).$$ Intuitively, $$K: [L^2(0,1)]^n ...
Gustave's user avatar
  • 545
0 votes
1 answer
140 views

Expectation of random matrix

Assume $Q$ is a positive definite random matrix such that $0 < \lambda_{\min}(Q)....\leq \lambda_{\max}(Q) \leq 1$ holds. I want to show that \begin{align} E\left[\frac{\lambda_{\min}(Q)}{\lambda_{\...
Ripon's user avatar
  • 3
3 votes
1 answer
312 views

What are the John ellipsoids for a pair of (9- and 15-dimensional) convex sets of $4 \times 4$ positive-definite matrices?

What are the John ellipsoids (JohnEllipsoid) for the 9- and 15-dimensional convex sets ($A,B$) of $4 \times 4$ positive-definite, trace-1 symmetric (Hermitian) matrices (in quantum-information ...
Paul B. Slater's user avatar
6 votes
1 answer
274 views

Recover approximate monotonicity of induced norms

Let $A$ some square matrix with real entries. Take any norm $\|\cdot\|$ consistent with a vector norm. Gelfand's formula tells us that $\rho(A) = \lim_{n \rightarrow \infty} \|A^n\|^{1/n}$. Moreover, ...
ippiki-ookami's user avatar
4 votes
1 answer
107 views

Is the Loewner maximum uniquely defined?

Given 2 (symmetric) PSD matrices $A,B$, is the following set $S_{A,B}$ non-empty? $$ S_{A,B} = \{ C: C\succeq A, C\succeq B, \text{ and }\forall D, D\succeq A, D\succeq B \implies D\succeq C \} $$ If ...
user163087's user avatar
0 votes
1 answer
141 views

Is it a sufficient condition for linearity?

Edit: According to the comment by LSpice we come back to the initial version of this question Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a smooth function such that for every $x\in \mathbb{R}^n$ the ...
Ali Taghavi's user avatar
4 votes
3 answers
231 views

Properties of matrix $X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}$

Let $\{\alpha_i\}_{i=1}^n$ be complex numbers such that $|\alpha_i|<1$, and consider the following $n\times n$ structured matrix $$ X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}. $$ Such ...
Ludwig's user avatar
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1 vote
1 answer
112 views

Asymptotic behavior of a matrix equation and its eigenvalues

We have a matrix valued function $A:\mathbb{R}_+\to \mathbb{R}^{m\times m}$. It is known that $A(\lambda)$ is a positive definite matrix for all $\lambda\in\mathbb{R}_+$ Denoting $\rho_i(A(\lambda))$ ...
Rajesh D's user avatar
  • 704
8 votes
1 answer
1k views

Closed form solution for $XAX^{T}=B$

Let $d \times d$ matrices $A, B$ be positive definite. Is there a closed form solution for the following quadratic equation in $X$? $$X A X^{T} = B$$ Thank you.
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