Questions tagged [matrices]

Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.

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1 vote
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94 views

Inequality on matrix trace

Consider the following inequality of Lemma 1 arising in The law of large numbers for quantum stochastic filtering and control of many-particle systems : $$\Big|tr(L\gamma LB) - \frac{1}{2}tr(B(L\gamma ...
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5 votes
2 answers
329 views

Trace identity for $2 \times 2$ reflections

Let $A, B, C \in \mathrm{GL}(2,\mathbb{C})$ be reflections (i.e., their eigenvalues are $\pm 1$). Please show that $$ \DeclareMathOperator\Tr{Tr}\{\Tr(AB)\}^2+\{\Tr(BC)\}^2 + \{\Tr(CA)\}^2 - \{\Tr(AB)\...
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1 vote
0 answers
108 views

Equivalent definition of positive semidefinite

I am reading a paper which repetitively uses the following statement, but I don't know why this is true: Statement Let $A$ be a symmetric $n$-by-$n$ matrix, and $B$ be a $n$-by-$n$ matrix, $A\geq 0$ ...
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7 votes
1 answer
221 views

Construction of skew-Hadamard matrix of order 292

I am currently looking into how to construct a skew-Hadamard matrix of order 292. Where can I find such construction? According to multiple papers (e.g. Koukouvinos and Stylianou - On skew-Hadamard ...
1 vote
1 answer
30 views

Control the summation of a diagonal matrix and another matrix to be full rank

Statement. To ensure the rank of $\operatorname{ddiag}(AQQ^T)-\sigma\Delta=n$, it is sufficient to require $\min_i(\operatorname{diag}(AQQ^T))_i>\sigma\lVert\Delta\rVert$. Note: $Q\in\mathbb{R}_{n\...
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0 votes
0 answers
85 views

On the exponentiation of a stochastic matrix where the exponent is a function of matrix size

In this question, I asked about any arbitrary stochastic matrix $A(n)$ of the particular form $$A(n) = \begin{pmatrix} 1 & 0 & \cdots & 0\\ x_{21} & x_{22} & \cdots & 0\\ \...
0 votes
0 answers
39 views

A question on the paper "On the low-rank approach for semidefinite programs arising in synchronization and community detection"

I got stuck at theorem 15 when reading this paper "On the low-rank approach for semidefinite programs arising in synchronization and community detection" by Bandeira, Boumal, Voroninski. ...
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1 vote
2 answers
42 views

Monotonicity of kernel matrices with respect to hyperparameters

Let $\mathcal{X}$ be some nice space, let $\Phi$ be some ordered space, and let $K :\mathcal{X} \times \mathcal{X} \times \Phi \to \mathbf{R}$ be a positive-semidefinite kernel indexed by a ...
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1 vote
0 answers
42 views

What do you call this class of matrices with a unique positive eigenvalue associated to a graph?

I am looking for the name of a class of symmetric matrices $M\in\Bbb R^{n\times n}$ that I can associate to a (finite simple) graph $G=(V,E)$ with $V=\{1,...,n\}$ and that have the following ...
  • 11.1k
3 votes
0 answers
80 views

Solvability of a matrix exponential equation - generalized matrix logarithm

For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation $$ G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) . $$ Basic ...
1 vote
1 answer
53 views

Convergent condition of the high-dimensional submatrix of some orthogonal matrix

Let $\mathbf{V}$ be a $p\times p$ orthogonal matrix (i.e., $\mathbf{V}\mathbf{V}^\top = \mathbf{V}^\top \mathbf{V} = \mathbf{I}$) whose columns are $$ \mathbf{V} = \begin{bmatrix} \mathbf{v}_1 & \...
4 votes
1 answer
133 views

largest eigenvalue of the difference between two quadratic forms

Let $U,V\in\mathbb{R}^{4\times n}$ such that $UU^T=VV^T=I$, and $A\in\mathbb{R}^{n\times n}$ be an Hermitian matrix. Is it true that $$\sqrt{\lambda_{\text{max}}\left(\left(UAU^T-VAV^T\right)^2\right)}...
  • 505
1 vote
0 answers
15 views

eigenvalues and eigenvectors of K blocks matrix

I'm trying to find the eigenvalues and eigenvectors of the following $n\times n$ matrix, with $k$ blocks. $$X = \left( \begin{array}{cc} A & B &... & \\ B & A & B & ... \\ B &...
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0 votes
0 answers
28 views

Testing a condition in linear algebra involving Krylov subspaces

Let $A \in \mathcal{M}_n(\mathbb{R})$ be a real-valued $n \times n$ matrix. For $b \in \mathbb{R}^n$, I consider the Krylov subspace $$K_A(b) = \operatorname{span} \{ b, A b, \dotsc, A^{n-1} b \}.$$ ...
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6 votes
1 answer
289 views

Is the set of purely real square matrices, that are complex-diagonalisable, dense in the set of real matrices?

A quick search for "diagonalisable matrix" on Wikipedia immediately gives the result that the set of real-diagonalisable matrices is not dense in the set of real matrices. I need, however, ...
1 vote
1 answer
40 views

Eigenvalues of a circulant: DFT or Inverse DFT Convention?

Currently, most engineering texts (and webpages including Wikipedia) define forward discrete Fourier transform with a negative sign on the exponential. This is a convention and the inverse discrete ...
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3 votes
1 answer
151 views

Why is the square root of a complex symmetric matrix also complex symmetric

I am looking at trying to show that a complex symmetric matrix always has a complex symmetric square root. Showing a square root exists is fairly easy if the matrix is also invertible by using the ...
2 votes
1 answer
52 views

Generate a low-rank sparse covariance matrix

May I ask how to generate a low-rank sparse covariance matrix? Thank you!
-2 votes
0 answers
113 views

Property of positive semi-definite

Let $A$ is a positive semi-definite matrix like this: $$ A = \begin{bmatrix} 1 & \alpha_{1,2} & \alpha_{1,3} & \alpha_{1,4}\\ \alpha_{1,2} & 1 & \alpha_{2,3} & \alpha_{2,4}\\ \...
3 votes
1 answer
109 views

On the bounds of the sum of the squares of spectral variation of two real symmetric matrices

Suppose $A$ and $B$ are two symmetric real $\{0,1,-1\}$ matrices of order $n$ with diagonal elements as zeros (therefore the traces are zeros) and eigenvalues $\lambda_1\ge \lambda_2\ge \dotsb \ge \...
0 votes
0 answers
67 views

How to show two equivalent projection in a C∗ algebra are not homotopic

Show that two equivalent projections need not be homotopic. HINT: Let $P=\begin{pmatrix} 1&0\\0&0\end{pmatrix}$ and $Q=\begin{pmatrix} t&\sqrt{t(1-t)}\\\sqrt{t(1-t)}&1-t \...
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3 votes
2 answers
207 views

Extend an inequality on matrix norms

Let $A$ denote an $n \times n$ matrix, and $\sigma_i(\cdot)$ denote $i$-th largest singular value. Can we extend the following result to general $p \geq 1$? For all $k = 1, \dots, n$, $$ \sum_{i = 1}^...
5 votes
0 answers
285 views

Decomposition of a determinant

Let $M$ be a $4\times 4$ symmetric matrix whose entries $m_{i,j}$ for $i,j =1,\dots,4$ are homogeneous polynomials of degree $2$ in $3$ variables. Assume that $m_{1,1} = 0$. Does there exist a ...
  • 433
8 votes
7 answers
929 views

One observation of special type of square matrix exponentiation

I was studying the following type of matrices, $$ A = \begin{pmatrix} 1 & x_{12} & \cdots &x_{1n}\\ 0 & x_{22} & \cdots &x_{2n}\\ \vdots\\ 0&\cdots&0&x_{nn} \end{...
0 votes
0 answers
39 views

What information concerning the eigen-structure are transformed on the antidiagonal submatrices?

Let us fix symmetric matrices $A_1$ $A_2$ in $M_m(\mathbb{R})$ with $A^2_1=\alpha I$ and $A^2_2=\beta I$ for some positive $\alpha ,\beta$. For a given matrix $B\in M_m(\mathbb{R})$, let us ...
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4 votes
2 answers
791 views

English translation of “A multidimensional generalization of the Wronskian”

I am trying to generalize an algorithm for the construction of a certain linear combinations of functions $\boldsymbol{f}:x\in\mathbb{R}\mapsto \boldsymbol{f}(x)\in\mathbb{R}^n$ that utilizes ...
  • 11.7k
3 votes
0 answers
73 views

When does a multiplicative subset of matrices have positive trace?

Let $A_1, A_2,..., A_m$ be a collection of real $n \times n$ matrices. What are some conditions (necessary or sufficient) on $A_1,...,A_m$ for any product $A_{i_1} A_{i_2} \cdots A_{i_k}$ to have non-...
2 votes
0 answers
70 views

How to decompose a matrix over a ring $F[X_1,\ldots,X_k]$ as a product of two matrices

Let $F$ be a field. Assume any reasonable conditions if needed, such as $F=\mathbb R$, $F=\mathbb C$, $F$ is a finite field, or $F$ has a specific characteristic, etc. Let $C$ be an $n\times1$ matrix ...
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0 votes
0 answers
68 views

What are the properties of square-matrix algebra with this equivalence class?

Consider the set of all square matrices with the following equivalence class: $\mathbf{A}\sim\mathbf{A}\otimes\mathbf{I}_n$ (or, alternatively, as user @M.G. proposed, $\mathbf{A}\sim\mathbf{I}_n\...
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1 vote
2 answers
81 views

Transforming matrix to off-diagonal form

I wonder if one can write the following matrix in the form $A = \begin{pmatrix} 0 & B \\ B^* & 0 \end{pmatrix}.$ The matrix I have is of the form $$ C = \begin{pmatrix} 0 & a & b & ...
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2 votes
0 answers
47 views

any ideas on how to solve matrix equation like this $X A_i Y = B_i$

the objective function is like $$\operatorname*{argmin}_{X,Y} = \sum_i \|X A_i Y - B_i\|_F^2$$, and $A_i$ is a diagonal matrix I've tried gradient-descent, but as it turns out not well, I wonder if ...
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4 votes
0 answers
133 views

Algorithm for generalized Hilbert's Theorem 90 over $\Bbb R$

$\newcommand{\GL}{\operatorname{GL}} \newcommand{\R}{{\Bbb R}} \newcommand{\C}{{\Bbb C}} $For a natural number $n$, let $z\in \GL(n,\C)$ be a 1-cocycle of $G=\GL_{n,\R}\,$, that is, an invertible ...
1 vote
1 answer
97 views

eigenvalues of matrices (with positive entries)

I am reading an old paper by Kawpien and Pelczynski, Studia Math. 1970. It claims that singular values of a matrix (with positive entries? I am not sure) is given by $t_i=\sqrt{\sum_{j\ge 1}a(i,j)^2}$....
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3 votes
1 answer
249 views

Positive system of algebraic integers

Let $\mathbb{A}$ be the ring of algebraic integers. Consider a sequence $(d_i)_{i \in I}$, with $I$ a finite set and $d_i \in \mathbb{A} \cap \mathbb{R}_{\ge 1}$, such that $$d_i d_j = \sum_{k \in I} ...
3 votes
0 answers
127 views

Spectrum of large Hilbert matrices

Let $x_k>0$ be a increasing sequence of real numbers, such that $$\sum_0^\infty\frac1{x_k}<+\infty.$$ Let us form the (infinite) Hilbert matrix $A\in{\bf Sym}({\mathbb N};{\mathbb R})$ with $$a_{...
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2 votes
0 answers
50 views

Entropy of eigenvectors of a large matrix

My question pertains eigenvectors of matrices with somewhat evenly distributed entries. Let $M$ be an $N \times N$ matrix with complex entries (think of $N$ as a large integer). You can assume that $M$...
  • 2,554
0 votes
0 answers
76 views

Relationship between singular values, traces and Hermitian conjugate

I am working on a following problem in my free time (which is a simplified version of a problem described here - arxiv.org/abs/0711.2613): Let $A$, $B$ be zero-trace $4 \times 4$ matrices that meet ...
1 vote
0 answers
31 views

Enumerate all possible sign patterns spanned by matrix column space

Given a $m \times n$ matrix $A$ with $m>n$, I would like to enumerate all possible sign patterns $w$ generated by $Av$ for all $v \in \mathbb{R}^n$. More specifically, if $(Av)_i \geq 0$ then $w_i =...
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6 votes
1 answer
420 views

Conjecture on the existence of centrosymmetric Hadamard matrices

I work with centrosymmetric matrices and recently have started exploring the question of the existence of centrosymmetric Hadamard matrices. Definition: An $n \times m$ matrix $A = (a_{i,j})$ is ...
1 vote
0 answers
23 views

Bounds on Eigenvalues After Skew-Symmetric Perturbation

Consider two matrices $\mathbf{J} \in \mathbb{R}^{n \times n}$ and $\mathbf{L} = -\mathbf{L}^T \in \mathbb{R}^{n \times n}$. I am trying to upper bound the eigenvalues of their sum: $$\mathbf{A} = \...
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1 vote
0 answers
94 views

Kernel of a Vandermonde type matrix

Consider a complex matrix $A\in\mathbb{C}^{n+1\times m}$ such that $$ A=\begin{bmatrix} 1 & 1 & \dots & 1\\ Bc_1 & Bc_2 & \dots & Bc_{m} \end{bmatrix}, $$ where $c_1,\dots ,...
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1 vote
0 answers
46 views

Diagonal entries and determinant of positive definite matrices

Given a $3x3$ symmetric, positive matrix $\lambda |x|^2 \leq x^TAx \leq \Lambda |x|^2$, let us denote $a_{11}$, $a_{22}$ and $a_{33}$ to be the diagonal elements. Furthermore, let us denote the values ...
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3 votes
2 answers
184 views

Eigenvalues and eigenvectors of non-symmetrical tridiagonal matrix

The question is the following: given a matrix $$A=\begin{pmatrix} 1& 2 & & & & \\ 1& 0& 1 & & & \\ & 1& 0& 1 & &\\ & &...
3 votes
1 answer
131 views

Results on Boolean matrices

Matrices with entries in the finite field of two elements $\mathbb{F}_2$, and with the usual operations of matrix addition and multiplication, have been intensively studied, especially due to their ...
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1 vote
3 answers
197 views

How to optimize for the best fit nonuniform-scale-rotation to a given 3×3 matrix?

Given a matrix $L\in \mathbb{R}^{3 \times 3}$, I'm looking for a method to find the closest (in a least squares sense) product of a non-uniform scaling matrix and a rotation matrix: $$ \min_{s\in\...
3 votes
2 answers
407 views

Reducing $9\times9$ determinant to $3\times3$ determinant

Consider the $9\times 9$ matrix $$M = \begin{pmatrix} i e_3 \times{} & i & 0 \\ -i & 0 & -a \times{} \\ 0 & a \times{} & 0 \end{pmatrix}$$ for some vector $a \in \mathbb R^3$, ...
1 vote
1 answer
59 views

A question about the sign of quadratic forms on nonnegative vectors

Let $M$ be a real square matrix of order $n\ge 3$. Assume that for every nonnegative vector $\textbf{z}\in \mathbb R^n$ which has at lease one zero entry we have $\textbf{z}^T M \textbf{z} \ge 0$. Can ...
2 votes
1 answer
128 views

Existence of finite dimensional representation of an algebra

Let $m>1$ be an integer and let $A$ be the algebra generated by the elements $\{u^i_j,v^i_j,\bar{u}^i_j, \bar{v}^i_j| 1\leq i,j\leq m\}$ quotient over the relations \begin{eqnarray} u^i_j v^k_l&...
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9 votes
1 answer
302 views

One question on circulant $(-1,1)$-matrices

Let $n > 13$ be a positive integer. Is there any $n\times n$ circulant $(-1,1)$-matrix $A$ satisfying the following property: $$AA^T=(n-1)I+J$$ where $I$ is the $n\times n$ identity matrix and $J$ ...
1 vote
0 answers
124 views

How to maximize certain function of hundreds variables related to correlations between sets vectors ? (and win Kaggle :))

It might be helpful for data science/bioinformatics challenge. Consider for simplicity three rectangular matrices $Y_{true}$ , $Y_{predict0},Y_{predict1}$ of the same sizes say 70000*140. Let us ...

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