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 ...

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1answer
747 views

Conjugate Matrix

Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix ...
3
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0answers
41 views

When is a Hankel matrix invertible?

I would like to have some conditions that render a general Hankel matrix of the form \begin{pmatrix}a_1 & a_2 & a_3 &\cdots & a_n \\ a_2 & a_3 & a_4 & \cdots & a_{n+1} ...
-4
votes
0answers
44 views

how can I make a grid [on hold]

I have 16 numbers, 1 to 16. 16 rows. 4 columns. I want each number to appear in each column but to only appear on a row with each of the other 15 numbers once.
0
votes
2answers
111 views

Inverse of a matrix expression

Let $$X_i = \left(I - P\left(I - t_it_i^T\right)\right)^{-1}$$ where $P$ is an $N\times N$ matrix and $t_i$ is a vector of $N$ elements. Is there a way to simplify this expression in order to ...
1
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2answers
89 views

Matrices congruent to each other via a permutation

Consider the collection of all integer matrices and partition them via an equivalence relation $A\sim B\Leftrightarrow \exists$ a permutation matrix $P$ such that $B=PAP^T$. Is some canonical form ...
0
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1answer
90 views

$\mathbb{Z}_{2}$ -equivariant vector bundles over manifold of rank-$k$ matrices

Edit: I remove the trivial part of the first version, according to comment of Alex Degtyarev Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. There is ...
4
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1answer
123 views

Hankel matrix commuting with a Jacobi matrix

Assume the semi-infinite Hankel matrix $H$ with entries $$H_{i,j}=\alpha_{i+j-1}, \quad i,j\geq1,$$ where $\alpha_{n}\in\mathbb{R}$, is given. It turns out that in some special situations a ...
2
votes
1answer
111 views

Rank changes with matrix edits

Assume we have rank $r$ real matrix $M\in\{0,1\}^{n\times n}$ (not constrained to symmetric). Assume $W\in\{0,1\}^{n\times n}$ is rank $1$ real matrix. Case $1$: $M+W\in\{0,1\}^{n\times n}$. Could ...
0
votes
2answers
467 views

Similarity about unitary matrices

Suppose $G_1, \ldots, G_k$ are unitary, Hermitian, and anti-commuting matrices, and assume the same for $F_1, \ldots, F_k$. Suppose these matrices are similar, i.e. there exists $T \in ...
5
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2answers
651 views

Proving that the kernel of this matrix is of dimension 2

(Edit : see at the bottom of the question for an additional surprising possible hint.) Using a computational software program, I found that the kernel of the following matrix is of dimension 2 when ...
-4
votes
2answers
234 views

Representing quaternions as matrices [on hold]

Assume F is a field of characteristic different than 2. Let a, b be invertible elements in F, and let A(a,b) be the generalised quaternions. Using the Artin–Wedderburn theorem, there is a ...
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0answers
42 views

how to find coordinate of unknown point given the distance against N known points [migrated]

I am meeting with a problem, say I have already know the coordinates of N points (a1,a2,a3....) in 3D space. And I have a new point, say x. I only know the distances from x to the known N points. Is ...
3
votes
1answer
70 views

Probe permutationally matrix extreme properties

Suppose given $M\in\{0,1\}^{n\times n}$ of rank $r$. Assume that changing even a single $1$ to $0$ in $M$ raises rank. Does it follow that $M$ is permutationally equivalent to a block diagonal ...
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0answers
23 views

Probe permutationally matrix extreme properties-II

Call $S_{r}$, collection of $0/1$ matrices of rank atmost $r$ that increase rank if any $1$ is changed to $0$. Given $M\in\{0,1\}^{n\times n}$ of rank $r$, what is probability that $M$ could be ...
2
votes
1answer
345 views

Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$. First, let's define two matrices: ...
3
votes
1answer
209 views

Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements

Assume that $A$ is a unital $C^{*}$ algebra. Is there a subvector space $Y\subset A$ of finite codimension which does not contain any invertible element? Let $n(A)$ be the infimum of such ...
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0answers
34 views

Spectral radius of a column stochastic matrix perturbed by a rank-1 matrix

$P\in \mathbb{R}^{n\times n}$ is an irreducible column stochastic matrix. $P$ is also diagonally dominant. $w \in \mathbb{R}^{n} $ is a strictly positive vector satisfying $w^T \mathbf{1} = 1$ where ...
0
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0answers
56 views

A combinatorial question on ranks

Denote $$\mathscr{C}[r]=\{Q\in\Bbb Z_{\geq 0,\leq 1}^{n\times n}:\mathsf{rk}(Q)= r\}.$$ $$\mathscr{D}[A,t]=\{B\in\mathscr{C}[\mathsf{rk}(A)]:\mathsf{dim}(col(A)\cap col(B))\geq t\}.$$ Given ...
5
votes
0answers
140 views

Characterizing matrices with rank constraint

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M,b]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad ...
2
votes
1answer
55 views

Maximizing Frobenius Norm of Commutator (an opposite Procrustes problem)

I was wondering if anybody has any suggestions on the following problem: Let $S$ be an $n\times n$ positive definite symmetric matrix. I wish to find an $n\times n$ orthogonal matrix $R$ which ...
2
votes
1answer
137 views

Symplectic (contact) structure on $M_{n}(\mathbb{R})$

Assume that $n$ is an even number. What is a natural symplectic structure on $M_{n}(\mathbb{R})$, such that for every $1\leq k \leq n$ the manifold of $k$-rank matrices would be invariant under ...
3
votes
1answer
82 views

Upper bounds on elements of a matrix

During my research I have come across matrices this type $$C=B\left(B^T B\right)^{-1}B^T\ ,$$ where $B$ is an $m\times n$ real matrix. If $B^TB$ is not invertible, then $\left(B^T B\right)^{-1}$ ...
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0answers
26 views

Linear combinations of columns of matrices [closed]

Suppose E is a 4 × 3 matrix with columns ⃗c1, ⃗c2, ⃗c3 and rows ⃗r1, ⃗r2, ⃗r3, ⃗r4. Let ⃗v be a 3 x 1 matrix that = [2, -1, 1] How could we express E⃗v as a linear combination of ⃗c1, ⃗c2, ⃗c3? Now ...
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2answers
1k views

Eigenvalues of nonnegative integer matrices

Edit I realized that the key piece of information that I need is question 1, and so I'd like to rephrase this post: What are the possible eigenvalues of nonnegative integer matrices? Any answer ...
2
votes
2answers
170 views

A line bundle over the manifold of singular matrices

According to answer of Denis Serre to this question, the manifold of singular matrices in $M_{n}(\mathbb{R})$ is defined as follows: $$M=\{A\in M_{n}(\mathbb{R})\mid \text{rank}(A)=n-1\}$$ So we ...
3
votes
1answer
95 views

A norm description for singular matrices

For $n>2$, are there norms $\parallel.\parallel_{a}$ and $\parallel.\parallel_{b}$ on $M_{n}(\mathbb{R})$ with the following property: $A\in M_{n}(\mathbb{R})$ is singular if and only if ...
4
votes
1answer
111 views

Non-negative decomposition of a non-negative matrix

Consider a matrix $A\in{\bf M}_{n\times m}({\mathbb R})$, whose entries are non-negative. Let $r$ be the rank of $A$. It is well-known that $A$ decomposes as $x_1y_1^T+\cdots+x_ry_r^T$ with ...
127
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19answers
17k views

Geometric Interpretation of Trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandry; Is there a geometric interpretation of the trace of a matrix? This question ...
2
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1answer
262 views

Diagonalization of 4th order tensors

I have been wondering about the following problem... Let $n$ be a positive integer and denote by $M_n^s$ the space of symmetric $n\times n$ real matrices. Now, we look at the space $\mathcal ...
11
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2answers
594 views

Hölder's inequality for matrices

I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if $$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$ But ...
0
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1answer
139 views

A geometric property of singular matrices

Let $S\subset M_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$. That is $S$ is the critical points of the determinant function. What matrices belongs to $S$, precisely? Let ...
1
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0answers
69 views

Spectrum of primitive nonnegative integer matrices

Let $P(X) = a_nX^n + \cdots + a_1X + a_0$ with $a_i \in \mathbb Z$. Question 1. Is there an efficient criterion on the $a_i$ to decide if there exists a primitive nonnegative integer matrix with ...
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0answers
98 views

Matrices with a common Fischer basis

Let $A$ be a real symmetric $n\times n$ matrix, normalized such that $Tr[A]=1$. Define a 'Fischer basis' as the basis in which all diagonal elements are equal to $\frac{1}{n}$. The motivation for ...
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0answers
30 views

Bound of spectral radius of polynomial of a complex matrix

I am trying to prove or disprove the following inequality. $$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$ where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and ...
2
votes
2answers
149 views

Matrices with real spectrum

Assume you have a non-symmetric real square matrix of all whose eigenvalues are real. Can anything be said about it? Is it unitarily equivalent to a symmetric matrix? EDIT: Is it at least similar to ...
1
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0answers
96 views

Eigenvalue of product of self adjoint compact operators

Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
3
votes
3answers
311 views

Are the finite dimensional von Neumann algebras, singly generated?

Let $\mathcal{M}$ be a finite dimensional von Neumann algebra, then : $$\mathcal{M} \simeq \bigoplus_i M_{n_i}(\mathbb{C})$$ Question : Is it singly generated (as von Neumann algebra)? how ? ...
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0answers
110 views

A quantity associated with an algebraic variete

Let $P:\mathbb{C}^{n}\to \mathbb{C}$ be an irreducible homogenous polynomial. Is there a geometric or algebra geometric interpretation for the following quantity: The maximum number $k$ such that ...
8
votes
1answer
136 views

Is the center of the automorphism group of a von Neumann algebra M trivial whenever M is a factor?

Question: Is the center of the automorphism group of a von Neumann algebra $\mathscr{M}$ trivial (=$\{\mathrm{id}\}$) whenever $\mathscr{M}$ is a factor (=$\mathscr{M}$ has center $\{\lambda I; ...
1
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1answer
98 views

Matrix Submodular Inequality

Given $a,b,x > 0$ I know following the submodularity property holds: \begin{align} \frac{1}{a} - \frac{1}{a+x} \geq \frac{1}{a+b} - \frac{1}{a+b+x} \end{align} My question is, does this property ...
0
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1answer
146 views

Bounded domain matrix factorization

Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$. The goal is to return $\widehat A, \widehat B$ such ...
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2answers
90 views

About adding a negative definite rank-1 matrix to a symmetric matrix

If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$) I guess that the eigenvalues of $B - vv^T$ ...
11
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0answers
224 views

Why is a matrix pencil called a pencil?

I'm trying to understand the historical context behind the word pencil in matrix pencils, or pencil of curves so on. I am aware that even Gantmacher 1959 has this terminology however I don't know ...
6
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1answer
220 views

Jordan decomposition of the tensor product of two matrices

I asked this question on Math.SE here, but did not get a lot of attention. I am interested in the problem of determining the Jordan decomposition of the tensor product of two unipotent matrices over ...
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0answers
37 views

Orthogonalization technique after cosparse dictionary update

I'm trying to adapt the cosparse dictionary learning (DL) approach described in Analysis K-SVD to a DL method that creates the dictionary as a union of orthonormal blocks (UONB). For this I apply the ...
0
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0answers
39 views

Dual cone of a set of particular semidefinite cones

Let $X$ be a matrix variable $$X=\begin{pmatrix} x_1 & x_2 & x_3\\ x_2 & x_4 & x_5\\x_3 & x_5 & x_6\end{pmatrix},$$ define the cone as ...
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0answers
72 views

An exact fraction of a matrix

Let $A$ be a $n \times m$ real matrix with $n<<m$ and of rank $r<n$. It is known that $A$ has exactly two distinct non-zero singular values: $\sigma_{\max}$ and $\sigma_{2}$, and also that ...
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0answers
36 views

Bits of precision matrix reconstruction

We have a real rank $r$ matrix $M\in\{0,1\}^{n\times n}$. Suppose we have diagonalized using $LMR=D$. I want to recover a real matrix $\widetilde{M}$ such that maximum absolute entry of ...
5
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1answer
414 views

Some calculus in the orthogonal group $O(n)$

How can one compute each of the following matrices, explicitly: $$\int_{O(n)} e^{g}dg$$ or $$\int_{O(n)} g^{n}dg \;\;\;\;n\in \mathbb{N} \;\;n>1$$ What is the explicite entries of the resulting ...
0
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0answers
28 views

Way to parameterise sparse multi diagonal matrix

I have an NxN matrix S that looks like this: $$ S^{-1} = K^{-1} + \Lambda $$ where N is a multiple of 3, both K and S are positive definite matrices, and Lambda is $$ \Lambda = \begin{bmatrix} x ...