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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98 views

Schrodinger operator with matrix potential

This question is inspired by attempts to extend in some way Shen's 1999 "On fundamental solutions of generalized Schrödinger operators" to Schrödinger operators $- \Delta + V $ with some ...
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34 views

Schaten p norm of block matrices

Let $A=D\oplus 0$ be a diagonal Hermitian matrix and $B$ is an invertible Hermitian matrix with $(1,1)$ block being $B_{11}$ and $B_{11}$ and $D$ have the same dimensions. Then is it true that if $(1+|...
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40 views

Find occurrences of certain matrix inside a matrix

This problem occurred from my need to find all graphs with a certain topology inside a bigger one. I don't need the subgraphs but the graphs that have the exact topology I am searching. We know for ...
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1answer
51 views

Bound for matrix inner product based on singular values

Regarding the matrix inner product based on singular values, Lewis (1995) "The convex analysis of unitarily invariant matrix functions" states the result by von Neumann that $\langle X,Y \rangle \leq \...
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80 views

Upper and lower bounds on the entries of a matrix power

Say I have a non-negative square $n\times n$ irreducible stochastic matrix $A$ (i.e., each column sums to 1), for which the following holds: $$A_{ij} > 0 \iff A_{ji} > 0.$$ I know that no more ...
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1answer
272 views

Inequality for $AB + BA$ when $A,B\geq0$, reference request

Let $A,B\geq0$ be positive semidefinite matrices of arbitrary size $n\times n$. Denote by $\alpha$ and $\beta$ their largest eigenvalues. It is well-known that the eigenvalues of the expression $AB +...
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34 views

Minimum rank of a product of two block diagonal matrices with an arbitrary matrix

Let us assume that we have an arbitrary full-rank $l\cdot b \times l\cdot p$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), an $m \times ...
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69 views

Random sparse and invertible matrices

Let $n\leq m$ and $0\leq k\leq (n\times m - \min\{n,m\})$ be in $\mathbb{N}$. Let $\mu$ be a probability measure dominated by the Lebesgue measure on $\mathbb{R}$ and generate a random $n\times m$ ...
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46 views

A system of inequalities involving a skew-symmetric integer matrix

Which skew-symmetric integer matrices $S$ satisfy the following inequalities $SV_i \ne z_iE_i$ for all $i = 1,\cdots, n$ where $V_i$ denotes the column with integer entries such that the $i$-th ...
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149 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 $m\times (n\cdot l)$ ...
2
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0answers
57 views

Convergence rate of Toda/Morse flow

Let $A(t), A_0$ be a $n\times n$ hermitian complex matrices and consider the following matrix flow \begin{align} \frac{dA}{dt} &= \left [ C\circ A , A \right ] \\ A(0) &= A_0 \ . \end{align} ...
6
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1answer
95 views

Minors of low rank skew-symmetric matrix

Let $A$ be an $n\times n$ skew-symmetric matrix of rank $r$. Given subsets $X$ and $Y$ of row and column indices respectively, let $A_{X,Y}$ denote the submatrix of $A$ obtained by only keeping rows ...
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55 views

Matrix Lie algebra: Semisimple element = semisimple matrix?

I am considering the set $$ \mathbb{L} = \lbrace X \in \mathbb{C}^{2n \times 2n} \; ; \; J^{-1}X^HJ = -X \rbrace, \quad J = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix},$$ of complex ...
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1answer
121 views

Joint convexity of trace of matrices

Let $\Gamma_{m\times m}$ be a diagonal matrix with positive diagonal entries and $\mathbf{A}_{m\times m}$ be an arbitrary matrix. Then, is the following trace function jointly convex on $\Gamma_{m\...
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11 views

Update rank-revealing factorization after applying $I \otimes $(small matrix)

Suppose one knows an $(m, n)$ matrix $A$ along with some rank-revealing factorization (let's say the economy SVD for conreteness), $A = U S V^H$, with $S$ an $(r, r)$ matrix where $r$ is the numerical ...
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34 views

Spectral abscissa of symmetric matrix with skew-symmetric perturbation

I am interested in bounds on the minimal distance between the spectral abscissa $\max_{\lambda\in\sigma(A)}\mathrm{Re}\lambda$ of a matrix $A$ and the eigenvalues of its perturbated version $A+S$. In ...
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3answers
536 views

Representation theorem for matrices (reference request)

Motivation. If $A \in \mathbb{C}^{n \times n}$ is self-adjoint (or, more generally, normal), then we all know that $$ A = \sum_{k=1}^n \lambda_k \, h_k \otimes h_k, $$ where $\lambda_1,\dots,\lambda_n$...
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533 views

Spectral norm bound on smooth primary matrix function perturbation

Consider an $L$-Lipschitz function $f: \mathbb{R} \to \mathbb{R}$ (so $|f(x) - f(y)| \leq L|x-y|$ for all $x,y$) and Hermitian PSD matrices $A, B \in \mathbb{C}^{n\times n}$. Define $f(A)$ to be $f$ ...
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1answer
138 views

Eigenvalues of product of symmetric positive definite matrices

Let $T_1, \ldots, T_n$ by real symmetric positive definite matrices, with eigenvalues bounded below by $\mu > 0$. Can I say $$ \frac{x^T T_1 T_2 \ldots T_n x}{x^T x} \geq \mu^n $$ If these matrices ...
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1answer
67 views

Choosing the best submatrix

Let $\mathbf{A}_{m\times n}$ be a matrix with non-negative elements. Assume that a submatrix $\mathbf{B}$ from $\mathbf{A}$ is defined as \begin{align} B_{i,j} = \begin{cases} A_{i,j}, & i\in\...
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1answer
130 views

LU decomposition for orthogonal or unitary matrices?

Is there any references on LU decomposition for orthogonal or unitary matrices? It seems to me that the diagonal entries of $U$ has some nice structure regarding to the Euler angles of the original ...
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60 views

Eigenvalues of the matrix obtained by letting some of the rows vanish, hoping for some inequality

Let $A$ be an $n \times n$ matrix. Let $A_k$ be the matrix obtained by keeping the first $k$ rows of $A$ fixed and substituting $0$ for the rows $k+1$ to $n$. To be precise, we write $A= [R_1...R_k, ...
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1answer
580 views

What are known properties of matrices where off-diagonal elements are 1?

Consider a matrix where the diagonal entries are anything but the off-diagonal entries are all one. I was able to find a formula for the determinant of this matrix, but what are other known properties?...
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54 views

On norm convergence of a commutator and implications

Given a sequence of real symmetric $(2^N\times 2^N)$- matrices $(H_N)_{N\in\mathbb{N}}$ on the $N$-fold tensor product of $\mathbb{C}^2$ with itself, such that \begin{align} \lim_{N\to\infty}||[H_N,...
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2answers
693 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 ...
4
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1answer
275 views

Can I assign the term “is eigenvector” and “is eigenmatrix” of matrix **P** in my specific (infinite-size) case?

remark: I asked this in MSE, the question got views and votes but seemingly no one had an answer so far. Background: I'm rereading a couple of my exploratory (surely not research-...
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0answers
148 views

Matrix equations motivated by generalization of $QR$ decomposition

The following problem is motivated by considering a certain generalization of the $QR$ decomposition of a matrix. Let $A, B \in M_n(\mathbb{R})$. (i) Can we always find $Q_1, Q_2 \in M_n(\mathbb{R})...
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1answer
534 views

Eigenvectors of Kronecker Product [closed]

Conjecture If $A$ and $B$ are two complex square matrices, then every eigenvector of $A\otimes B$ is of the form $x\otimes y$, where $x$ is an eigenvector of $A$ and $y$ is an eigenvector of $B$. ...
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55 views

Nullity of infinite matrix in row echelon form

For an $m \times n$ matrix $A$ in row echelon form, $\mathrm{nullity} (A)$ is equal to the number of columns that do not contain a pivot. Is this also true for an infinite matrix in row echelon form ...
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75 views

Symmetrization of pentadiagonal matrices

Nonsymmetric tridiagonal matrices $T_3$ can easily be symmetrized via a (diagonal) similarity transformation $D=\text{diag}(d_1, \dots, d_n)$ (i.e. see Wikipedia) $$ J_3=D^{-1} T_3 D \,. $$ Is there ...
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2answers
309 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 ...
28
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2answers
4k views

Consequences of eigenvector-eigenvalue formula found by studying neutrinos

This article describes the discovery by three physicists, Stephen Parke of Fermi National Accelerator Laboratory, Xining Zhang of the University of Chicago, and Peter Denton of Brookhaven National ...
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0answers
114 views

Transformations preserving the number of distinct eigenvalues

Let $A\in\mathbb{R}^{n\times n}$ be an $n\times n$ symmetric, invertible matrix with nonnegative real entries, $\mathbf{1}$ be the all one $n$-dimensional vector, and $\mathrm{diag}(v)$, $v=[v_1,v_2,\...
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0answers
92 views

Controlling the rank of a Matrix product

Let $\bf{Q}$ be an $(m-k)\times m$ matrix satisfying $\bf{QQ}^H=\bf{I}$, $\bf{W}$ an $m\times m\ell$ matrix of the form $$\bf{}W=\left[\begin{array}{ccccc}{\bf{w}}_1 &\bf{0} &\bf{0}&\...
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0answers
95 views

Existence of solution for system of quadratic equations

I have a system of quadratic equations $${\bf{x}}_m {\bf{A}}_{k} {\bf{y}}_k^{\mathrm{H}}= Z_{m,k}, \quad 1\leq m \leq M,\;\; 1\leq k \leq K,\;\; M>K $$ where ${\bf{x}}_m$ is $1\times N$, and ${\bf{...
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1answer
140 views

How to derive the mean of inverse-Wishart distribution?

How to derive the mean of inverse-Wishart distribution in Inverse-Wishart distribution? I have no idea about it. Thanks for your help.
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55 views

What is the expected inverse of 1 plus a Wishart distribution?

Let $$ X \sim \mathcal{W}_{n} (n, \Sigma) \; \;$$ Where $\mathcal{W}$ denotes the Wishart distribution and $\Sigma \in \mathbb{R}^{q \times q}$ is the corresponding scale matrix (symmetric, positive ...
6
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1answer
284 views

How wide is the Birkhoff Polytope?

This question is migrated from MSE where it turned out to be much harder than I thought. I still cannot figure this out. Does anyone have any ideas? Define the width of a polytope $P \subset \mathbb ...
3
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0answers
250 views

Homotopicity of $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ as morphisms from $A$ to $A\otimes A$

let $A$ be a $C^*$ algebra. We equip $A\otimes A$ with the spatial norm. Assume that two morphisms $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ are homotopic morphisms, i.e, there is a curve $\...
6
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1answer
710 views

Quantum inspired matrix inequality

While mimicking the union bound in quantum systems, we land on the following conjecture but don't know how to prove this. Given any complex-valued $n\times m$ matrix $A$. A sub-matrix of $A$ is ...
4
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1answer
125 views

Row-based iterative algorithms for computing the kernel of a matrix

Suppose $A$ is an $m \times n$ matrix in the form $$A=\begin{pmatrix} — a_1 —\\ — a_2 —\\ \vdots \\ — a_m — \end{pmatrix}$$ where $a_i \in R^n$ is the $i$-th row of $A$. I know that it is possible ...
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1answer
40 views

Checking a matrix for distinct rows

I am computing some combinatorial parameter associated with the complex simple Lie algebras of type $A_n$ using sage and the output will be a rectangular integer matrix. All I need to prove to solve ...
3
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1answer
251 views

Differentiability of operator norm [closed]

Is there any known results about differentiability properties of the function $\mathbb f:\mathbb R \to\mathbb R,$ $f(t):=\|A+tB\|_{op}$ where $\|.\|_{op}$ denotes the usual operator norm of the ...
2
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0answers
41 views

Conjugation of Irreducible {0,1}-Matrices by Orthogonal Matrix

Let $A_1$ and $A_2$ be irreducible matrices with entries in $\{0,1\}$. Let $S$ be an orthogonal matrix such that, $A_1=SA_2S^T$ Is it necessarily the case that $S$ is a permutation matrix?
5
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2answers
132 views

Coloring in Combinatorial Design Generalizing Latin Square

I have a question about a combinatorial design very similar to a Latin Square, which is arising out of an open problem in graph theory. The design is an $n \times n$ matrix whose entries we want to ...
2
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1answer
64 views

Additivity of the Field of Values

If $A \in M_n(\mathbb{C})$, then the field (of values), or numerical range of A, is the compact, convex subset of the complex-plane defined by $$ F(A)= \{z^* A z \mid z^*z = 1 \}. $$ It is well-...
4
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2answers
137 views

Frobenius normal form of a doubly stochastic matrix

If $A \in M_n(\mathbb{C})$, then $A$ is called reducible if there is a permuation matrix $P$ such that $$ P^\top A P = \begin{bmatrix} A_{11} & A_{12} \\ 0 & A_{22} \end{bmatrix}, $$ in ...
0
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0answers
63 views

Jordan Decomposition of Sparse matrix

Suppose we are given $n \times n$ rational matrix, $A$ with at most $k$ nonzero elements in each row and each column with $k \ll n$. What is the best algorithm to compute its Jordan decomposition? ...
5
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1answer
209 views

Best orthogonal approximation of rank 1 matrix

Let $X=\lambda_0u_0v_0^T\in\mathbb{R}^{n\times n}$ be a rank 1 matrix where $\lambda_0\in\mathbb{R}$, $u_0,v_0$ are of unit Euclidean norm. What is the solution of the following problem? $$\hat{X}=\...
4
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0answers
258 views

Do we know what the impulse to “introduce” the Jordan canonical form was?

Mo-ers, Do you know how it was that the study of the Jordan canonical form began? There are certain things that may be said once one has thought about the matter: for instance, one can say that the ...

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