Questions tagged [matrix-theory]

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

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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 ...
xmonetx's user avatar
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6 votes
0 answers
481 views

concentration for eigenvectors

I am interested in bounding from above the ratio between the maximum and minimum entries of a Perron vector. The only results that I found in the literature are from the classic masters (Ostrowski and ...
Felix Goldberg's user avatar
5 votes
1 answer
643 views

Is every real matrix conjugate to a semi antisymmetric matrix?

Is it true to say that every matrix $A\in M_n(\mathbb{R})$ is similar (conjugate) to a matrix $B=(b_{ij})$ with $b_{ij}=-b_{ji}$ for all $i\neq j$?(With some abuse of terminology,a matrix $B$ with ...
Ali Taghavi's user avatar
40 votes
4 answers
4k views

The sum of squared logarithms conjecture

I am searching for the first proof of (or counterexample to) the following conjecture. (The sum of squared logarithms conjecture) For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , ...
Patrizio Neff's user avatar
28 votes
2 answers
15k views

Determinants in Graph Theory

In graph theory, we work with adjacency matrices which define the connections between the vertices. These matrices have various linear-algebraic properties. For example, their trace can be calculated (...
Ion Georgiou's user avatar
16 votes
5 answers
2k views

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 ...
Hipstpaka's user avatar
  • 355
4 votes
0 answers
1k 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 ...
wonderich's user avatar
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3 votes
1 answer
339 views

Maximal common isotropic subspace for a finite family of skewforms

Let $V$ be a vector space of dimension $n$ over a field $F$. An alternating bilinear form $\alpha\colon V \times V \rightarrow F $ will be called a skewform. A subspace $W$ is isotropic for $\alpha$...
Sky's user avatar
  • 913
1 vote
0 answers
351 views

Asymptotic expansion square root matrix

I am looking for an asymptotic expansion for $\underline\gamma$ which is the "square root" matrix of a symmetric $p\times p$ matrix $\gamma$. Here $\underline\gamma$ is assumed to be symmetric, e.g. ...
user68601's user avatar
32 votes
0 answers
624 views

Existence of orthogonal basis of symmetric $n\times n$ matrices, where each matrix is unitary?

For a positive integer $n$, let $S_n$ denote the set of $n\times n$ symmetric matrices over $\mathbb{C}$. As a complex vector space, this set has dimension $\mathrm{dim}(S_n)=\binom{n+1}{2}$. The ...
Mark Girard's user avatar
18 votes
2 answers
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Minimum off-diagonal elements of a matrix with fixed eigenvalues

I am an engineer working in radar research. I came accross a problem on which I cannot seem to find literature. I can ask it in two different ways. Perhaps depending on the reader, the alternative ...
mermeladeK's user avatar
17 votes
1 answer
3k views

2x2 subdeterminants of a matrix

If N>2, it is well known that if two invertible NxN matrices A and B have the same determinants of any 2x2 corresponding submatrices, then A=B or A=-B. Given then all these 2x2 determinants of an ...
Carlo Mantegazza's user avatar
17 votes
2 answers
970 views

Are unitarily equivalent permutation matrices permutation similar?

Two matrices $A, B \in \mathbb R^{n \times n}$ are called unitarily equivalent if there exists an unitary matrix $U \in \mathbb C^{n \times n}$ such that $A = U B U^{\ast}$. If in addition $U$ is a ...
shuhalo's user avatar
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15 votes
7 answers
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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 ...
Adam's user avatar
  • 345
13 votes
2 answers
5k views

Parametrization of positive semidefinite matrices

We know that a real, symmetric, positive definite matrix $A$ of size $n\times n$ can be parametrized by a vector $\theta$ of $\frac{n(n+1)}{2}$ parameters thanks to the Cholesky decomposition: $$ A = ...
epsilone's user avatar
  • 313
13 votes
2 answers
390 views

Is every finite-order unimodular matrix conjugate to a $0,1,-1$ matrix?

Problem. Given a matrix $A\in\mathrm{GL}(n,\mathbb{Z})$ such that $A^k=1$ for some $k\geq 1$, is there a matrix $g\in\mathrm{GL}(n,\mathbb{Z})$ such that $gAg^{-1}$ has only $0$, $1$, and $-1$ as ...
Qfwfq's user avatar
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12 votes
2 answers
20k views

Cholesky decomposition of a positive semi-definite

We know that a positive definite matrix has a Cholesky decomposition,but I want to know how a Cholesky decomposition can be done for positive semi-definite matrices?The following sentences come from a ...
Purple's user avatar
  • 121
11 votes
1 answer
427 views

A variant of Cholesky decomposition involving binary matrices

Studying a problem that is not directly related to linear algebra I came across the following problem. Let $B$ be $n \times n$ symmetric matrix whose entries are non-negative integers. I would like ...
Jernej's user avatar
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10 votes
2 answers
5k 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}(\|...
Hans's user avatar
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9 votes
3 answers
538 views

Product of a Finite Number of Matrices Related to Roots of Unity

Does anyone have an idea how to prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} x^{-2j} & -x^{2j+1} \\ 1 & 0 \end{pmatrix}\right)= \begin{cases} ...
David Sun's user avatar
  • 309
8 votes
1 answer
1k 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 ...
messcode's user avatar
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.
Fabio's user avatar
  • 329
8 votes
0 answers
177 views

Rational points with small denominator in $U(n)$

Fix integers $n,d>0$. (I'm probably thinking about $n\leq 6$ and $d\leq 2000$.) Let $X$ be the set of matrices $A\in U(n)$ such that the entries of $dA$ lie in $\mathbb{Z}[i]$. Is there an ...
Neil Strickland's user avatar
6 votes
0 answers
197 views

Conjecture for a certain Cauchy-type determinant

Given the Cauchy-like matrix $$ \mathbf X_M(q) = \left[ \frac{2}{\pi} \frac{ \Gamma\!\left(m - \frac{1}{2} \right)\Gamma\!\left(n + \frac{1}{2} \right) }{ \Gamma(m)\,\Gamma(n) } \frac{m-\frac{3}{4}} {\...
Fred Hucht's user avatar
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6 votes
0 answers
206 views

Reference request: maximal determinant of matrices with pairwise orthogonal rows and entries in $\{1, 0, -1\}$

We know that "Hadamard maximal determinant problem" concerns the largest determinant of a matrix of oder $n$ with entries in $\{-1,1\}$ or $\{0, 1\}$. For $n=4k$, it is the Hadamard ...
Arun 's user avatar
  • 725
5 votes
4 answers
2k views

Differentiability of eigenvalue and eigenvector on the non-simple case

Let $h:\mathbb{R}^n\to\mathbb{R}^m, n>1$ be a twice continuously differentiable function and $J_h:\mathbb{R}^n\to\mathbb{R}^{m\times n}$ be its jacobian matrix. Let us consider the functions $A(x):=...
Shake Baby's user avatar
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5 votes
2 answers
2k views

Decomposing a matrix into a product of sparse matrices

How to study the decomposition of a square matrix into a product of sparse matrices? There are no restrictions on the number of matrices in the product, but the fewer the better.
unknown's user avatar
  • 451
4 votes
2 answers
307 views

Solving $\text{trace}\left[\left(I + pY\right)^{-1} \left(I - p^{2}Y\right)\right] = 0$ for scalar $p$

For given $n\times n$ real symmetric positive definite matrices $X_{1}, X_{2}$, let $Y := X_{1}^{-1}X_{2}$, and let $I$ be the identity matrix. I would like to solve the following equation for the ...
Abhishek Halder's user avatar
4 votes
1 answer
612 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 ...
Math_Y's user avatar
  • 311
4 votes
1 answer
289 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? ...
Turbo's user avatar
  • 13.6k
4 votes
0 answers
2k views

eigenvalues of a symmetric tridiagonal matrix with zero diagonals

I was investigating a problem and came up with the following symmetric tridiagonal matrix (with zero diagonal elements): $$ \left(\begin{array}{cccccc} 0 & a & 0 & \ldots & 0 \\ a &...
Xiao Junhui's user avatar
3 votes
0 answers
228 views

Multi-dimensional permanent of structured tensor

I am facing the multidimensional permanent \begin{equation} \text{perm}(W) = \sum_{\sigma, \rho \in S_n} \prod_{j=1}^n W_{j, \sigma_j, \rho_j } \end{equation} of a 3-tensor $W_{j,k,l}$ of ...
Malte's user avatar
  • 93
3 votes
1 answer
685 views

Regular semisimple elements in $SL(n,q)$

Consider $G=GL(n,q)$. A regular semisimple element of this group, is a matrix, whose characterestic polynomial is square-free and same as minimal polynomial. Results show that the number of such ...
Rijubrata's user avatar
  • 117
2 votes
0 answers
165 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 ...
Heinrich A's user avatar
2 votes
2 answers
300 views

The structure of the $n$-th power of a special matrix

Assume the following matrix $$ C_p^{(a,b)}:=\left( \begin{array}{cccccc} a &a &0 &\cdots &\cdots &0 \\ 0 &0 &a &\ddots &\ddots &\vdots \\ \vdots &\ddots &...
Amin235's user avatar
  • 325
2 votes
1 answer
109 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^...
Hadi Asheri's user avatar
2 votes
0 answers
164 views

Lower bound for the sum of cosines between singular vectors of diagonally dominant matrices

Let $A \in \mathbb{R}^{n \times n}$ be a nonsymmetric diagonally dominant matrix with $a_{ij} < 0$ $\forall i \ne j$ and $a_{ii}>0$. Let the singular value decomposition of $A$ be $A=U \Sigma V^...
Astor's user avatar
  • 323
2 votes
0 answers
169 views

Minimum of $\mathrm{rank}\left( \boldsymbol{W} \boldsymbol{H} \right)$, with $\boldsymbol{W}$ block diagonal

Let us assume that we have a full-rank $(n\cdot l)\times k$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), and an $m\times (n\cdot l)$ ...
Juan's user avatar
  • 61
1 vote
1 answer
125 views

Redistribute diagonal entries of a matrix

Let $d = (d_1,...,d_k)^t$ with positive entries. Denote $D:=diag(d)$ and let $m > k$. What are sufficient conditions on $d$ and $m$ so that there exists $V \in \mathbb{R}^{m \times k}$ with: $V$ ...
Yair Daon's user avatar
  • 185
1 vote
0 answers
63 views

Does there exist a canonical form for normal matrices which extends the following embedding?

Given an unordered pair of complex numbers $\{w,z\}$, we can associate to it the complex matrix $$\frac 1 2\left[\begin{matrix}w + z + \frac{\left(w - z\right)^{2} + \left|{w - z}\right|^{2}}{2 \left|{...
wlad's user avatar
  • 4,792
0 votes
1 answer
110 views

A conjugation matrix $X\in \mathbb{C}^{ n\times p}$ where $p< n$

Given a Hermitian positive definite matrix $A\in \mathbb{C}^{n \times n}$ and a Hermitian matrix $B\in\mathbb{C}^{ p\times p},$ find the matrix $X$ so that $X^HAX=B$ holds where $X^H$ denotes ...
Saheb's user avatar
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