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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2-positivity to 3-positivity

Let $B\in M_3(\mathbb{C})$ and $S_3= \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{pmatrix} $. Suppose $I_3\otimes I_2+B\otimes X+B^*\otimes X^*\geq 0$ for all $2$...
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Minimize $Q(t,s) = \mathbf{w}^\top \begin{pmatrix} r(t, t) & -r(t, s) \\ -r(t,s) & r(s, s) \end{pmatrix}^{-1} \mathbf{w}$ in $(t, s)$

Let $0 \leq r(t,s) \leq 1$, $t, \ s \in [0, T]$ be a smooth enough function, such that $r(t,t)$ increases in $t$ $r(t, s) = r(s, t)$ decreases as $t$ and $s$ move away from each other (that is, as $|...
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Proof that sum of k-eigenvalues is convex

I saw a post A sum of eigenvalues that said It is well-known that $\sum^{r}_{i = 1} \lambda_{i}(X)$ is convex. and I saw an explanation in Boyd and Vandenberghe - "Convex Optimization", ...
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What are the convergence requirements for Inverse Power Method?

I'm struggling to find the convergence requirements for the Inverse Power Method. I implemented this method in MATLAB as shown below. ...
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When is the Fourier discrete matrix almost similar to some particular diagonal matrix?

Let $N$ be a natural number and put $z=e^{\frac{\pi i}{N}}$ and $w=z^2$. Let us consider the discrete Fourier matrix $F=(w^{kl})_{k,l=0,\cdots,N-1}$ and the diagonal matrix $D=\operatorname{diag}(1,...
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Adding the AWGN to the data makes its covariance matrix always positive definite?

I'm working on a numerical method that estimates direction-of-arrivals in antenna arrays. I realized that every time I add the AWGN (Additive white Gaussian noise) to a data (which is a matrix), its (...
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2 votes
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On the eigen vectors of a diagonalizable matrix

Let us consider the space $M_n(\mathbb{C})$. By a unitary matrix $U=(u_{ij})$ we mean that $U^{-1}=(\overline{u_{ji}})$. Q. Let $U$ be a unitary matrix. I am looking for the pairs of matrices $(D,A)$ ...
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10 votes
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362 views

(Asymmetric) matrix power series in closed form: $\sum_{i=0}^{\infty} A^i \left(A^i\right)^{\top}={?}$

Let $A\in \mathbb{S}^{N\times N}$ be a symmetric, real and stable matrix, i.e., $\rho(A)<1$, where $\rho(A)$ stands for the spectral radius of $A$. Then, $$\sum\limits_{i=0}^{\infty} A^{2i}=\left( ...
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4 votes
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Total positivity tests: optimal in the number of minors vs. the computational cost

A real matrix is called totally positive if all of its minors are positive. Of course, to check that a given matrix is totally positive it is not necessary to check all the minors: for example, it ...
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Define a matrix square root that preserves regularity

Let $A:\mathbb{R}\to \mathbb{R}^{n\times m}$ and $B\in \mathbb{R}^{n\times k}$. Is it possible to define $C:\mathbb{R}\to \mathbb{R}^{n\times m}$ satisfying the following two properties: for all $t\...
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Determinant Inequality with unitary matrix

I come up with the following conjecture while doing my research, which is a determinant inequality. I have tried to run the MatLab simulation to verify its sanity. It seems that the inequality is true....
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Factorization of an invertible matrix $A$ into $A=MB$ with $M$ symplectic

A matrix $M \in \mathbb{R}^{2n\times 2n}$ is called symplectic if $M^TJM=I$ and $J$ is the standard symplectic matrix $$ J = \begin{pmatrix} 0 & -I \\ I & 0 \end{pmatrix} \in \mathbb{R}^{2n\...
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How sparse can a matrix mapping between sparse vectors be?

Let $s$ and $d$ be non-negative integers with $0\leq s<d$ and let $v,u\in \mathbb{R}^d$ be vectors satisfying the sparsity estimate $$ \max\{\|u\|_0,\|v\|_0\}\leq d-s, $$ where, as usual, for any ...
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Are more duality mappings for matrix norms known?

When reading A Unifying Representer Theorem for Inverse Problems and Machine Learning by Michael Unser and Duality Mapping for Schatten Matrix Norms by his PhD student Shayan Aziznejad, I wondered if ...
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Principal component analysis with boundedness constraints

Let $A$ be an $m\times n$ matrix with entries in $F$ ($F=\mathbb{R}$ or $F=\mathbb{C}$). It is well-known that $A$ has decompositions of the form $$\displaystyle A = \sum_{k=1}^r\lambda_k\hspace{2mm} ...
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Banach-Mazur distance between Schatten-$p$ classes

Let $M_n$ denote the set of all $n\times n$ complex matrices. Let $1\leq p<\infty.$ For $A\in M_n$ define $\|A\|_p:=(Tr(A^*A)^{p/2})^{1/p}$ where $Tr$ denotes the usual trace of a matrix. Then $\|.\...
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Diagonal of the difference of two orthogonal matrices who are close to the identity matrix

Suppose that $\mathbf{A} = \mathbf{I}_{n} + \mathbf{E}_{1}$ and $\mathbf{B} = \mathbf{I}_{n} + \mathbf{E}_{2}$ are two orthogonal matrices where $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ are two small ...
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2 answers
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Inequality for matrix with rows summing to 1

Let $A$ be real matrix with $M > 1$ rows and $K > 2$ columns, and each entry $a_{m,k} \in (0,1)$, with each row summing to $1$. For all $m$ $$ \sum_{k=1}^{K} a_{m,k} = 1 $$ I want to find out if ...
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Unitarily equivalent matrices that are also unitarily equivalent on orthogonal subspaces

Consider two positive semidefinite matrices $A$ and $B$ on $\mathbb C^d$. Let $\{P_i\}_{i=1}^m$ be a complete family of $m$ orthogonal projectors on $\mathbb C^d$ (i.e., $P_i^*=P_i, P_iP_j=\delta_{ij}...
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Log determinant of quadratic form

I am reading a paper Cook and Forzani - Likelihood-Based Sufficient Dimension Reduction where the author uses the following result from matrix analysis but does not explain why it is true nor provide ...
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217 views

Singular value decomposition of truncated discrete Fourier transform matrix

Let $\mathbf{F}$ be a discrete Fourier transform (DFT) matrix such that \begin{align} F_{m,n}=e^{-j2\pi(m-1)(n-1)/N},\quad m,n=1,\ldots,N. \end{align} What we can say about the singular value ...
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Cofactor an geometrical mean in $\mathit{SPD}_3$: a Gårding-like inequality

The cofactor map $A\mapsto\widehat A$ is polynomial homogeneous of degree $n-1$ over $\mathbf M_n({\mathbb R})$. It can be polarized into an $(n-1)$-linear symmetric map. When $n=3$, this provides a ...
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3 votes
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Schur-Horn theorem for principal submatrices

The Schur-Horn theorem says that there exists a Hermitian matrix with diagonal entries $d_1,d_2,\ldots,d_n$ and eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_n$ if and only if $(\lambda_1,\lambda_2,\...
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1 vote
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Number of non-singular matrices with entries in $\{1, -1\}$

What is the number of non-singular $n \times n$ matrices with entries in $\{1, -1\}$? Here I assume that two such matrices are equivalent if one can be obtained from the other by permutations of rows ...
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Lower bound of the smallest singular value of Hadamard power

Suppose that $A \in \mathbb{R}^{m \times n}$, does the lower bound of the smallest singular value of hadamard power of $A$ ($A^{\circ 2} = A \circ A, A^{\circ 3} = A \circ A \circ A, \cdots$) exist? ...
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Fastest way for certain rectangle matrix multiplication

I have $2$ matrices over $\mathbb{N}$, from the size $n \times \sqrt{n}$ and $\sqrt{n} \times n$. I would like to find an efficient way to multiply them. By efficient, I mean better than $n^{2.5}$, ...
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1 answer
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Second order matrix differential equation in the space of symmetric positive definite matrices

In the construction of interpolations in the space of Gaussian measures, I encountered a second order matrix differential equation in the set of symmetric positive definite matrices $\mathbb{S}_+^d\...
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Problems concerning the existence of matrices with specific conditions

I'd like to collect in this thread questions about the existence of matrices in general size fulfilling specific criteria. I have been thinking whether it would be a good idea to create a specific tag ...
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5 votes
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Analysis proof of dual number spectral theorem

Let the dual numbers be $\mathbb R[\varepsilon]/(\varepsilon^2)$. Write a general dual number as $a + b \varepsilon$ (where $\varepsilon^2 = 0$). Given a symmetric matrix $M$ over the dual numbers (i....
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2 votes
1 answer
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Limitation through the singular values

Given matrix $X \in \mathbb{R}^{m\times n}$ and sequence $\left\{X^k\right\}_k$ converges to $X$ according to the Frobenius norm. I wonder that $\sigma_i(X^k)$ converge $\sigma_i(X)$ or not (where $\...
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Extension of the Gershgorin circle theorem for symmetric matrices and localization of positive eigenvalues

In mathematics, the Gershgorin circle theorem can be used to localize eigenvalues of a matrix (including symmetric). Let $A$ be a real symmetry $n × n$ matrix, with entries $a_{ij}$. For $i∈{1,…,n}$ ...
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Is there a sufficient and necessary condition for the factorization of the matrix $M_{4\times 4}$ into $A_{2\times 2} \otimes B_{2\times 2}$?

Is there a sufficient and necessary condition for matrix $M_{4\times 4}$, so that $M$ can be decomposed into the form of the $A\otimes B$?
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Largest nuclear norm of $n \times n$ symmetric matrices whose entries are between -1 and 1

Let $\cal M$ be the set of real symmetric $n \times n$ matrices whose entries are all in the interval $[-1, 1]$. I'm interested in understanding the largest possible nuclear norm of these matrices as ...
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3 votes
1 answer
97 views

Proof of Levinson-Durbin algorithm

Is there any article or reference book with a full proof of the Levinson-Durbin algorithm used for solving linear system with a Toeplitz matrix ?
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eigenvalues of the product of a unitary with a diagonal

In $M_n(\mathbb{C})$, suppose $U$ and $D$ are a unitary and an invertible diagonal matrix with eigenvalues $\{e^{i\theta_1},\cdots,e^{i\theta_n}\}$ and $\{e^{i\eta_1},\cdots,e^{i\eta_n}\}$ ...
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3 votes
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Matrix logarithm of unitary factor from polar decomposition of product of positive definite matrices

This question is crossposted from Math Stackexchange here. I crosspost without much edits as I think this is the best way to phrase the question and because I received no feedback on the original post ...
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1 vote
1 answer
113 views

How do the singular values of a Hankel matrix, generated by some data time series, change when we add/remove rows and columns?

Suppose I have a smooth time series $C(t)$ defined on the interval $t=[0,T]$, from which I extract the sub-series $c=\{x_1,\cdots,x_N\}$ of $N$ entries, where $x_i=C(i*T/N)$. Naturally, the number $N$ ...
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Does fraction-free Gaussian elimination use fractional row operations?

I would like to understand whether Gaussian elimination of an integer matrix, which uses only row operations of the form Addition (or subtraction) of row $i$ to row $j$ can be performed in ...
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Exponential decay of a random matrix falling into a ball

Let $A=U\Sigma V^T\in\mathbb{R}^{n\times n}$ be a random matrix defined in the following way: $U,V$ are uniformly distributed on the orthogonal group $O(n)$, $\Sigma$ is a diagonal matrix such that ...
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2 votes
1 answer
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Matrix-valued cumulant generating function for Wishart matrices

Suppose we have an axis-aligned Gaussian vector $v \sim \mathcal{N}(\mu, \sigma^2 I_{d \times d})$, and consider the Wishart matrix $W = vv^\top$. Is there a simple closed form/"Lowener order ...
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1 vote
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There is an observation on the eigenvalues of the sum of a kind of special Hermitian matrices. How to prove it?

Suppose $A$ and $B$ belong to a kind of special hermitian matrices, which have the following properties: $A$ and $B$ contain only one negative eigenvalue. the negative eigenvalue and the second-...
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12 votes
4 answers
635 views

Show that the eigenvalues of a non-symmetric matrix built from positive matrices have positive real parts

Let $A, B, C \in \mathbb{R}^{n\times n}$ such that $N = \begin{bmatrix} A & B\\ B^{\top} & C\end{bmatrix}$ is a symmetric positive definite matrix. I'm trying to show that the following matrix ...
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Proof $(AB)^{\top}=B^{\top}A^{\top}$ via $(AB)^{-1}=B^{-1}A^{-1}$ and density argument. Unification of $(AB)^{\star}=B^{\star}A^{\star}$ [closed]

Question: $\mathrm{cl}\, \mathrm{O}_n(\mathbb{R})=\mathrm{GL}_n(\mathbb{R})$? Context: Let $\mathrm{GL}_n(\mathbb{R})$ be a set of nonzero determinant real matrices dimension $n\times n$, and $\mathrm{...
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7 votes
1 answer
432 views

Is it true that $\lVert A\rVert \leq \lVert A^2\rVert$ for $A\in \operatorname{SL}(2, \mathbb{R})$ when $\operatorname{trace}(A)>2$?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\trace{trace}$Let $A \in \SL(2,\mathbb{R})$ and $\trace(A)>2$. Is it true that $$\lVert A\rVert \leq \lVert A^2\rVert,$$ where $\lVert \rVert$ is the ...
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2 votes
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Bounding eigenvalue/eigenspace perturbations for hermitian matrices

Let $H$ be a Hermitian $n \times n$ matrix. Let $V$ be another such matrix. For real $t$, let us consider the one-parameter family $$ H(t) = H + t V$$ of Hermitian matrices. Kato's perturbation theory ...
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Conditions for a block matrix to be Hurwitz stable

Consider the following block matrix: $$ A = \begin{bmatrix} 0 & I\\ -M & -I \end{bmatrix} $$ Suppose matrix $M$ is positive definite and satisfies $M\succeq \alpha I$, where $\alpha>0$ is a ...
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1 vote
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182 views

Trace minimization for generalized eigenvalue problem

In [1], it is shown in theorem 1.2 that for symmetric $n \times n$ matrices $A$, $B$, we have $$ \min_{Y \in Y^*} \text{tr}(Y^TAY) = \text{tr}(X^TAX) = \sum_{i=1}^p \lambda_i, $$ with $$ \text{ $X^...
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1 vote
1 answer
83 views

Matrix equation with projection matrix

I need to solve the following equation for $P \in\mathbb{R}^{r\times d}$ $$P - G_1G_2(\lambda P^\top(PAP^\top)^{-1}P + A^{-1} ) = 0,$$ where the other quantities are known: $A\in\mathbb{R}^{d\times d}$...
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1 vote
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Minimax type principle for a self-adjoint operator acting on a Hilbert space

Let $T\in\mathcal{B}(\mathcal{H})$ be a self-adjoint operator acting on a Hilbert space $\mathcal{H}$. Suppose $k\in\mathbb{N}$. Define $$\lambda_k(T)=\sup\limits_{\substack{\mathcal{M}\subseteq \...
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Unit modulus manifolds

I am conducting my research in optimization in manifold. I come up with the following question. Let $\mathcal{F}_{p,q}$ is the collection of all matrices $\mathbf{F} \in \mathbb{C}^{p \times q}$ such ...
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