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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Classification of elements $GL(d, \mathbb{R})$

Any $SL(2, \mathbb{R})$ is either elliptic or hyperbolic, or parabolic up to conjugacy; see here. Do we have the same classification for $GL(d, \mathbb{R})$? If so, could you please introduce some ...
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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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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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1 vote
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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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3 votes
2 answers
133 views

Solving linear matrix equation

Given matrices $A, B, C' \in \Bbb R^{2 \times 6}$, where $'$ denotes matrix transposition, and matrix $L \in \Bbb R^{2 \times 2}$, how can one solve the following linear matrix equation in $X \in \Bbb ...
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5 votes
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130 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 ...
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Why do these polynomials split almost in the middle?

Start with a palindromic sequence of integers $(a_0, a_1, \ldots, a_{n+1})$, i.e. $a_j=a_{n+1-j}$, and put $a_j:=0$ for $j<0$ and $j>n+1$. You may readily guess that the choice of the binomial ...
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8 votes
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Cardinality of the maximum points of the determinant on matrices with entries in [-1, 1]

By multilinearity, the maximum of the determinant on matrices with entries in the interval [-1, 1] is attained at a {-1, 1}-matrix. By the following example, the maximum is attained at infinitely many ...
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Relation between the dimension of vector spaces and dimension of the space [closed]

Let $A \in \mathrm{GL}(d, \mathbb{R})$ be an irreducible matrix. Assume that $\{V_{n}\}_{n\in \mathbb{N}}$ is a non-zero proper subspace $\mathbb{R}^d$ with dimension $t<d,$ such that $AV_{n}=V_{n+...
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Recovering eigenvalues of a matrix from its $p$th compound matrix

This question was motivated by Find the determinant of a matrix given the determinant of all $p\times p$ sub-matrices?. Let $A$ be an $n\times n$ matrix over a field. Suppose we are given the $p$th ...
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2 votes
1 answer
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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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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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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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1 vote
1 answer
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Special type of normal form of matrix in principal ideal domain

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}$I want to ask the following, Given $X \in n \times n$ matrix that all the elements are integers and $X=X^{T}$ is symmetric. Can one always ...
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Perturbation theory for $UV^*$ in singular value decomposition

There is a fair amount of research into perturbation theory for singular value decompositions (e.g. Liu et al's 'First-Order Perturbation Analysis of Singular Vectors in Singular Value Decomposition' ...
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Finding a basis for the range of a linear function

I realize this question is not high level but I have posted it on Math Stackexchange: Stackexchange question and have received some upvotes but no answers or comments, so I am trying here. I will need ...
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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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Expressing a diagonal matrix over a division ring

Let $D$ be a division ring and $\mathrm{GL}_n(D)$ the general linear group over $D$. It is not difficult to show that every $A\in\mathrm{GL}_n(D)$ can be written in the form $A=D(x)B$ where $D(x)=\...
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Product of two involutions in $\mathrm{PSL}_2(D)$

Let $D$ be a division ring and $\mathrm{PSL}_2(D)$. Suppose that $\overline{A}\in\mathrm{PSL}_2(D)$ where $A\in \mathrm{SL}_2(D)$. If $\overline{A}$ is identity, then $\overline{A}$ can express two ...
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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|{...
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7 votes
1 answer
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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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1 vote
1 answer
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Upper bound of rank of a matrix

Given a set $U=\{1,\ldots,c\}$, we have $t$-dimensional vector $v_i=(x_1,x_2,\ldots,x_t)$, where $x_j$ is a subset of $U$. We can check that there are $2^{ct}$ above vectors. Then we construct the ...
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Sequence in matrices converging to Identity

Let $M_{k}(\mathbb{C})$ be the set of $k \times k$ complex matrices. I am trying to find a sequence of polynomials $P_{n}: M_{k}(\mathbb{C}) \to M_{k}(\mathbb{C})$ (or continuous functions $f_{n}$) ...
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2 votes
1 answer
126 views

Existence of matrices with some invertibility properties

Prove that there exists five matrices $B_i \in \mathbb{F}_2^{5\times 10}$, $i\in \{1,2,3,4,5\}$, such that any two $B_i$'s form an invertible matrix in $\mathbb{F}_2^{10\times 10}$. I am interested ...
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3 votes
1 answer
72 views

Multiplicative identity of determinant of multiplicative action of a polynomial on a quotient ring (companion matrices)

Let $A$ be a commutative ring with $f,g\in A[x]$ monics. Consider the $A$-linear endomorphism $\mu_g^{(f)}\in \mathrm{End}_A\tfrac{A[x]}{\langle f\rangle}$ given by multiplication by $g$. For monics $...
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Relation between optimizing determinant and optimizing the smallest eigen value

I have a question. Is there any relation between optimizing determinant and optimizing the smallest eigen value (lambda_min) of an n*n matrix?
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Can this fixed point theorem generalize to infinite structures?

Suppose that $n$ is a natural number, $X$ is a set, and $S\subseteq X^{2}$ is a subset such that if $x,y\in X$, then there is a unique tuple $(x_{0},\dots,x_{n})$ where $x_{0}=x,x_{n}=y$ and $(x_{i},...
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2 votes
1 answer
113 views

Minimal Laplacian spread of a graph

Laplacian spread of a graph is the difference among the largest and the second smallest Laplacian eigenvalue of the graph. Is there any result or conjecture that discusses about the graphs having ...
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1 vote
0 answers
80 views

In matrix product, differentiate one element with respect to another element

Background Consider a system (roughly) along the lines of those shown in Sims, C. A. (2002). Solving linear rational expectations models, where you have $$ AX_{t+1} = CX_t + M $$ where matrix $M$ is a ...
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3 votes
1 answer
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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 $\alpha$...
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Refinement of Hoffman-Wielandt inequality

We know that, for two $n\times m$ real matrices $A$ and $B$ we have ($\sigma_k(M)$ is the $k$-th singular value of $M$ in decreasing order) : $$\sum_{k=1}^{\min(n,m)}|\sigma_k(A+B) - \sigma_k(A)|^2 \...
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7 votes
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Is this "semi-tensor product" something recently invented? Are there other usages of it?

The context: I was reading a paper in which they used the following definition called "Semi-Tensor Product" (STP) or "Cheng" product (In honor to its "inventor" D. Cheng):...
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3 votes
1 answer
570 views

Eigenvalues of a block matrix with zero diagonal blocks

Suppose $A$ is a $k_1\times k_2$ matrix with real entries, $k_1<k_2$. Let $M$ be the matrix \begin{equation} M:=\begin{pmatrix} 0_{k_1} & A\\ A^\top & 0_{k_2} \end{pmatrix}, \end{equation} ...
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2 votes
1 answer
264 views

Routh-Hurwitz criterion for matrices

The Routh-Hurwitz criterion explicitly specifies a finite set of inequalities on the coefficients of a polynomial, necessary and sufficient that all zeros lie in the unit circle or in the left half ...
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3 votes
1 answer
163 views

Operator norm of difference of matrix decompositions

This question is in part related to a question that I have already posed. Say I have two symmetric positive definite matrices and their respective Cholesky decompositions $\mathbf{A} = \mathbf{L}_A \...
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2 votes
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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 ...
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8 votes
1 answer
243 views

Kulkarni-Nomizu square root of the Riemann tensor

Given a Riemann tensor $Riem$, what are conditions such that $Riem=B\star B$ for some bilinear symmetric form $B$, where $\star$ is the Kulkarni-Nomizu product? It follows from the proof of ...
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2 votes
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161 views

System of matrix equations

Problem definition: Let $x_i \in \mathbb{R}^d$ and $a_i \in [0,1]$, for all $i = 1,\dots, k$ (with $k\geq d$). Define $M(a) = \sum_{i = 1}^k a_i x_ix_i^T,$ and assume $M(a) \succ 0.$ Question: Is ...
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2 votes
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When does a matrix subspace contain a full rank matrix?

Cross-posted at Math SE Let $S\subseteq M_{n,m}(\mathbb{C})$ be a $d$-dimensional subspace of the space of $n\times m$ complex matrices (with $n\leq m$, say). I am interested in figuring out ...
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1 vote
1 answer
119 views

Existence of matrices in the field $\mathbb{F}_2$ with some invertibility properties

All the matrices in this statement are in the field $\mathbb{F}_2$. Let $I$ be the identity matrix of size $10 \times 10$ and let $e_1$, $e_2$, $\ldots$, $e_{10}$ denote its rows. For $i\in \{1,5 \}$, ...
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2 votes
1 answer
74 views

Existence of a matrix in $\mathbb{F}_2$ with some invertibility properties

All the matrices in this statement are in the field $\mathbb{F}_2$. Let $I$ be the identity matrix of size $10 \times 10$. What are all the possible $n$ ($\geq 6$) for which there exists a matrix $X$ ...
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1 vote
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204 views

Upper bound on the sum of the smallest non-zero eigenvalues

Let $\mathcal A := \{ A_1, A_2, \dots, A_n \} \subset \Bbb R^{d \times d}$ be a set of symmetric and positive semidefinite matrices. For a matrix $A_k \in \mathcal A$, denote its (real) eigenvalues by ...
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2 votes
0 answers
37 views

Does there always exist a(n uniform) polynomial that makes a positive definite symmetric matrix with polynomial entries into a sum of squares?

Suppose that I have a square and positive definite for every evaluation $x\in\mathbb{R}^{n}$ symmetric matrix $M(x)\in(\mathbb{R}[x])^{s\times s}.$ Does there always exist a polynomial $p(x)\in\...
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6 votes
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Expressing a polynomial as the determinant of a matrix of linear forms

I have heard that it's a well known result (in theoretical computer science?) that if we have a polynomial $p(t_1,\dots,t_n)$ over $\mathbb Q$, we can find matrices $M_0,\dots,M_n/\mathbb Q$ such that ...
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1 vote
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For which sets of linearly independent complex matrices does there exist a vector whose image set under the given matrices is linearly independent?

For which sets $\{A_i \}_{i=1}^n \subseteq M_d$ of linearly independent $d\times d$ complex matrices does there exist a vector $v\in \mathbb{C}^d$ such that $$\text{dim} [\text{span}\{A_1v, A_2v, \...
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1 vote
0 answers
53 views

Spectral theorem for symmetric real tensors

Is there a definition of eigenvalues that allows to use a spectral theorem? Let $\mathbf{T}$ be a real fully symmetric tensor of order $3$ and size $N$. Its components can be represented as $T_{ijk}\...
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Is it possible to reduce eigenvalues of tensors to an matrix eigenvalue problem?

Can we construct a larger matrix $M$ such that its eigenvalues are the same as the eigenvalues of a tensor $T$ of order 3? Let $\mathbf{T}$ be a fully symmetric tensor of order $3$ and size $N$. Its ...
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