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Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to [tag:linear-algebra]). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur ...

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A parametrization of a class of stable matrices

Let $A\in\mathbb{R}^{n\times n}$ be a non-derogatory matrix with real and strictly negative eigenvalues. Furthermore, for simplicity, suppose that $\mathrm{tr}(A)=-1$. Q. I'm wondering whether it ...
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537 views

Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ matrices?

(Disclaimer : I know very well that $SO(N)$ has a Lie algebra of dimension $N(N-1)/2$ etc. This absolutely not the point of my question.) To make my problem more understandable, I start with the ...
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92 views

Bounds of Procrustes problem

We denote $\|\cdot\|_F$ as the Frobenius norm of some matrix. We define $f: \mathbb{R}^{d\times r}\times\mathbb{R}^{d\times r} \rightarrow \mathbb{R}^{d\times r}$ as the following: \begin{align} f(A,B)...
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71 views

About product of PSD matrices

In Theorem 3 in this paper, https://core.ac.uk/download/pdf/82822897.pdf, ``On a product of positive semidefinite matrices, A.R. Meenakshi, C. Rajian, Linear Algebra and its Applications, Volume 295, ...
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127 views

Matrix Bernstein for spherical random variables

Theorem 4.1 in Tropp's Matrix Concentration Inequalities provides an exponential concentration inequality for the spectral norm of a matrix $Z = \sum_i \gamma_i B_i $, where $\gamma_i$ are an i.i.d. ...
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126 views

Clebsch–Gordan decomposition for $\mathrm{SU}(2)$, in indices

Let $\pi_m$, $m \geq 0$, be the unitary irreps of $\mathrm{SU}(2)$. The Clebsch–Gordan decomposition then gives that $$ \pi_m \otimes \pi_n = \bigoplus_{k=0}^{\min(m,n)}\pi_{m+n-2k}.$$ But suppose I ...
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62 views

Derivative of complex matrix pseudo inverse with respect to real and imaginary components

I have a complex non-square matrix $\mathbf{Y}\in\mathbb{C}^{n \times m}$ whose inverse I compute using the Moore-Penrose pseudo inverse, $\mathbf{Z}=\mathbf{Y^+}$. I am interested in evaluating the ...
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27 views

find linear approximation of non-linear matrix transform [closed]

I have a square matrix denoted as $A$ and an element-wise square operator $\sigma$, such that $\sigma(A)=a_{ij}^2$,$\forall i,j$, $a_{ij}$ is the ith row and jth column element of $A$. Is there exists ...
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69 views

Characterizing a subclass of row-orthogonal matrices

Let $O\in\mathbb{R}^{n\times m}$, $m>n$, be such that $O O^\top =I_n$. (Here $\bullet^\top$ denotes transposition and $I_n$ the $n\times n$ identity matrix.) Consider the following partition of $O$,...
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1answer
79 views

On ranks of matrices with tensor structure

Fix two $2^t$ length vector of form $p=\begin{bmatrix}u_1&v_1\end{bmatrix}\otimes\dots\otimes\begin{bmatrix}u_t&v_t\end{bmatrix}$ and $r=\begin{bmatrix}w_1&z_1\end{bmatrix}\otimes\dots\...
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69 views

Generalizing Autonne-Takagi factorization

Autonne-Takagi factorization (Léon Autonne (1915) and Teiji Takagi (1925)) says that: A complex symmetric matrix can be 'diagonalized' using a unitary matrix: If $A$ is a rank-$n$ complex symmetric ...
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1answer
114 views

Connections between eigenvectors after matrix multiplication

Suppose we have an M$\times$N complex matrix $H$ and its singular value decomposition $H=U\Lambda V^*$ and an N$\times$N covariance matrix $R_s$ with its eigendecomposition $R_s = U_s\Lambda_sU_s^*$. ...
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1answer
173 views

On a condition for a matrix sum to be zero

Let $\{Y_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices ($\mathrm{rank}(Y_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive definite ...
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1answer
168 views

Square root of a large sparse symmetric positive definite matrix

I am trying to calculate $$Y = A^{\frac 12} X$$ where $A$ is a very large and sparse positive definite matrix, say, $10^4 \times 10^4$. Matrix $X$ is known and, say, $10^4 \times 100$. Is there any ...
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1answer
173 views

An upper bound on the Jordan condition number of a matrix

The Jordan condition number of a matrix $A$ is defined to be $\min_{V}\kappa(V)$, where $V$ ranges over complex matrices that satisfy $A = VJV^{-1}$ for $J$ being the unique Jordan normal form matrix ...
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28 views

Classifications of the indefinite generalized Cartan matrix

I want to know that the present results about classifications of generalized indefinite Cartan matrices. I only have known that the classifications of hyperbolic matrces.
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1answer
319 views

Well known matrix inequality?

I suspect that the following matrix inequality is well known, but I can't find a reference or proof: Given $n \times n$ symmetric matrices $A,B$ such that $I_n \leq A,B$, is the following true? $${...
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3answers
764 views

Differentiability of Eigenvalues - Perturbation Theory

first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever ...
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1answer
387 views

Is it hard to decide whether a matrix is a square of another matrix?

According to the well-know quadratic residue (QR) theory over integers, we know that it is hard to decide whether a given integer $m\in\mathbb Z_N$ is a quadratic residue (i.e., a square of another ...
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2answers
250 views

Finding an adjacency matrix whose cube's diagonal is equal to a given vector

How can I find all binary matrices $A$ such that $A^3$ is a non-negative, integer square matrix and $$\mbox{diag}\left(A^3\right)=b$$ for some given vector $b$? Is there a way to characterize all ...
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326 views

Nuclear norm as minimum of Frobenius norm product

Nuclear, or trace, or Ky Fan, norm of a matrix is defined as the sum of the singular values of the matrix. It is claimed that $$ \|X\|_\sigma = \min_{UV^T=X} \|U\|\|V\| = \min_{UV^T=X} \frac{1}{2}(\|...
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197 views

A variant of Specht's Theorem using sum of elements (rather than trace) of complex matrices?

Let us first recall Specht's Theorem. Denote by $\text{Mat}_{\mathbb{C}}(n)$ the set of all $n\times n$ matrices over the complex field $\mathbb{C}$. Let $A$ be a matrix in $\text{Mat}_{\mathbb{C}}(n)$...
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1answer
116 views

An inequality regarding projection

Let $a, b \in \mathbb{R}^k$ be two normalized vectors such that $a^T b << 1$. Define matrix $C$ such that $[a, b, C]$ is full column rank, and let matrix $D$ be positive definite. Define ...
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5answers
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Expected value of determinant of simple infinite random matrix

Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where $$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$ I would like to ...
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63 views

Is there a usual technical term for this set-valued function associated to a zero-one matrix?

Let $m$ and $n$ be cardinals, and let $A\in \{0,1\}^{m\times n}$ be any zero-one matrix, considered as a function $A\colon m\times n\to\{0,1\}$. For every $j\in n$, let $A(\cdot,j)^{-1}(1):=\{i\in m\...
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2answers
172 views

Regarding minimal elementary generators for $GL(n, \mathbb{Z})$

I have a result concerning the minimal number of elementary generators (and by this I mean generators which are elementary matrices) for $GL(3, \mathbb{Z})$. I'm currently working on extending the ...
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1answer
149 views

Matrix continued fractions

I am aware of the classical continued fraction in the field of real numbers, but recently I have come across the term matrix continued fraction and when I checked on the internet there are varieties ...
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30 views

Lower bounds on eigenvalues of Lyapunov solutions

Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$ and let $X\in\mathbb{R}^{n\times n}$, $X=X^\top>0$ be the solution of the following Lyapunov algebraic equation $$ AX+XA^\top=-BB^\top....
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1answer
145 views

A Handbook of Matrix Factorizations

I am looking for a good collection of facts regarding the various types of matrix factorizations, something like a "Handbook of Matrix Factorizations" or a very-thorough review paper. I am hoping for ...
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1answer
452 views

Eigenvalues and eigenvectors of tridiagonal matrices

What can I say about the eigenvalues and eigenvectors of the tridiagonal matrix $T$ given as $T = \begin{pmatrix} a_1 & b_1 \\ c_1 & a_2 & b_2 \\ & c_2 & \ddots & \ddots \\ &...
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1answer
202 views

Matrix elements of exponential of tridiagonal matrices

Is there a way to compute one matrix element of the exponential of a tridiagonal matrix without having to compute the rest of the elements? Motivation: I'm trying to find the first passage time ...
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1answer
66 views

$A_{n \times m} D_{m \times m} A^T_{m \times n} + \alpha I_{n \times n}$

Assume that we have a matrix product of form $B=A_{n \times m} D_{m \times m} A^T_{m \times n} + \alpha I_{n \times n}$. $D$ is a positive diagonal matrix, $I$ is identity matrix, $\alpha>0$ and $m ...
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1answer
75 views

Equality or inequality for determinant of $A_{n \times m} D_{m \times m} A^T_{m \times n}$

Assume that we have a matrix product of form $B=A_{n \times m} D_{m \times m} A^T_{m \times n}$. $D$ is a positive diagonal matrix and $m > n$. Is there any equality or inequality over $|B|$, $|AA^...
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1answer
221 views

If $S$ is a nonsingular symmetric matrix over a number field and $D_k$ is its principal minor of order $k$, is $\frac{D_k}{D_{k-1}} > 0$ always true?

In Chapter II, Paragraph 4, Section 1 of F. R. Gantmacher, The Theory of Matrices. Vol. 1 English translation by K. A. Hirsch. Chelsea Publishing Company. 1959. SBN 8284-0131-4, the following ...
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2answers
323 views

Inverse of particular lower triangular matrix

I have an $n \times n$ lower triangular matrix $A$ where $$A_{i,j} = {\bf x}_i{\bf x}_j^H,\quad i>j$$ $$A_{ii}=1, \quad 1 \leq i \leq n,$$ and ${\bf x}_i$ is a $1 \times k$ (row) vector, where $...
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1answer
58 views

Proving symmetry of trace function of special matrix

Let $W = aI_{n\times n} + bJ_{n\times n}$, where $I$ is an identity matix, $J$ is the matrix of all ones, $a,b\in\mathbb{R}$ and a+b>0. Also, let $A = \mathbf{P} - \mathbf{p}\mathbf{p}^{T}$, where $\...
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1answer
163 views

A property of positive matrices

Let $\mathbb{S}^{dn} \ni X \succeq 0$ with $d,n \in \mathbb{N}$, where $X \succeq 0$ indicates that $X$ is positive semidefinite. Now partition $X$ into the block form \begin{gather} \begin{pmatrix} ...
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101 views

The inverse of sum of two positive matrices with almost orthogonal supports

I am interested to find an approximate formula for $$A (A+B)^{-1} A\ ,$$ for two positive matrices $A$ and $B$ whose supports are almost orthogonal. If the support of $A$ and $B$ are orthogonal ...
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74 views

Ruelle-Perron-Frobenius for continuous time

I'm looking for a proof (or a reference for it) of the following result: Let $L$ be a $n \times n$ $Q$-matrix, $p_0$ probability vector such that $L(p_0) = p_0$ and $V\colon \Omega \rightarrow \...
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1answer
92 views

Matrix inequality with arbitrary large ratios

Let $M = (m_{ij})$ be $n \times n$ symmetric positive definite matrix. Then it can be proven that $$ M^{1/2}A M^{1/2} \succeq M^{1/2}D M^{1/2}\succ 0 $$ so \begin{equation} \lambda_{\min}(M^{1/2}A M^...
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Is the determinant the only multiplicative matrix function? [closed]

Is there a matrix invariant or property that is multiplicative, i.e., $$f(AB) = f(A) f(B)$$ other than the determinant? In addition, some matrix norms are submultiplicative, but is there a ...
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98 views

Number of distinct rows and columns in a matrix with bounded number of entries

How many distinct rows and columns a real square matrix can have (at least in symmetric case) such that rank of matrix is $r$ and entries: are from $\{-b,-b+1,\dots,0,\dots,b-1,b\}$? are from $\{-b,-...
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155 views

A vanishing sum of symmetric matrices

Let $\{G_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices (i.e., $\mathrm{rank}(G_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive ...
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39 views

On ranks of specific products

Given $a=p^{r_1}m_1$ and $b=p^{r_2}m_2$ where $r_1,r_2\in\Bbb Z$, $a,b\in\Bbb Q$ and $p$ is coprime to numerator and denominator of both $m_1$ and $m_2$ and $r_1$ is $-\infty$ if $a=0$ and $r_2$ is $-\...
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85 views

What is this matrix decomposition called and does it exist always? - II

Given a rank $2r$ matrix $M\in\Bbb Q_{\geq0}^{n\times n}$ can we find two matrices $M_+\in\Bbb Q_{\geq0}^{n\times n}$ and $M_-\in\Bbb Q_{\geq0}^{n\times n}$ each of rank at most $r$ such that $M=M_+-...
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1answer
127 views

On sum of matrices

Suppose we have a matrix $M\in\Bbb Q^{n\times n}$ with no $0$ elements and we write as sum of two matrices $M_1$ and $M_2$ on following constraint. $M_{1,ij}=M_{ij}$ and $M_{2,ij}=0$ or $M_{1,ij}=0$ ...
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1answer
251 views

What is this matrix decomposition called and does it exist always?

Given a rank $2r$ matrix $M\in\Bbb Q^{n\times n}$ can we find two matrices $M_+\in\Bbb Q_{\geq0}^{n\times n}$ and $M_-\in\Bbb Q_{\geq0}^{n\times n}$ each of rank $r$ such that $M=M_+-M_-$ holds? ...
3
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1answer
227 views

Closed-form expression for differential of matrix function

Let $X$ be a real $n\times n$ positive semidefinite matrix of rank $m\le n$ and let $Y\in\mathbb{R}^{m\times n}$ be the unique matrix satisfying (i) $X=Y^\top Y$, and (ii) $Y\, [I\, |\, 0]^\top = L$ ...
3
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1answer
186 views

Completing the square of a matrix expression

Let $A,C\in\mathbb{R}^{m\times n}$, $n\ge m$, $B\in\mathbb{R}^{n\times m}$, and $P$ be a real positive definite $m\times m$ matrix. Denote by $\mathcal{S}^n$ the space of $n\times n$ real symmetric ...
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1answer
65 views

Is there any relation between weights in the eigenvector (corresponding to least eigenvalue) and the columns of a correlation matrix?

This problem is motivated from one of my pattern mining research projects. Any helpful suggestions will be highly appreciated. Consider an $n \times n$ correlation matrix A such that all the off-...