The matrix-theory tag has no wiki summary.

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### Bounds on smallest Eigenvalue of the Sum of a Standard Laplacian and a Diagonal Matrix

I'm trying to find upper boundaries on the smallest Eigenvalue $\lambda_1$ of $L + E$, where $L$ is a standard Laplacian of an unweighted digraph, with $\lambda_1(L) = 0$ and $E \in \{0,1\}^{n \times ...

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### Calculating the dimension of the algebra generated by some given matrices

Let $X_1, X_2, \ldots, X_d$ be $n \times n$ matrices over some field $K.$ I want to calculate the dimension of the unital algebra generated by $X_1, X_2, \ldots, X_d$ for some examples in a problem I ...

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### Distribution of the $\alpha$-parameter of a $2\times 2$ Haar-distributed, unitary matrix

It is well known that any $2\times 2$ unitary matrix $\mathbf{U}$ can be parametrized as
$$\mathbf{U}=\begin{pmatrix} 1 & 0 \\ 0 & \mathrm{e}^{\mathrm{j}\beta_1}\end{pmatrix} \begin{pmatrix} ...

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### Can anyone help me deduce a matrix inequality?

The following lemma is taken from references firstly.
Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$
for all $F$ satisfying ...

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### Questions about some special tensor transformation

Suppose tensor $U_{i\alpha\beta}$ with dimension $M*N*N$ satisfy following condition:
$$U_{i\beta\alpha}=W^1_{\alpha\alpha'}W^2_{\beta\beta'}U_{i\alpha'\beta'}$$
where $W^1$ and $W^2$ are $N*N$ ...

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### A problem that involves matrix and Lorentz Transformation [closed]

To be clear I address the question in two parts as below. All matrixes involved are real four-dimensional matrixes.
$1.$Let $G$ be the matrix $diag(1,-1,-1,-1)$. $A$ is a matrix satisfying $A G ...

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### On the positivity of matrices

For given $n$, the following $n\times n$ real matrix $M=M^{T}$ is called positive, if
$x^{T}M x\geq 0$
holds for all non-negative real $x_1,x_2,\cdots,x_n$,
where $x=(x_1,x_2,\cdots,x_n)^T$.
...

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### What are the upper bound and stability conditions for the following simple linear system?

Consider the following linear system
$$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1)
$$
where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...

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### 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 ...

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### Ergodic theory reference for converging sequences of matrices

I have been told that the following is a well known theorem in ergodic theory & have been given the book by Furstenberg as a reference. However, I cannot find such a statement in it. Would anyone ...

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### How to find matrix representations of a boolean algebra? [closed]

Given a boolean algebra with a finite number of elements {a, b, c, ...}, and the usual operations: $\cup, \cap, \neg$.
How to find matrix representations of the elements such that:
boolean $\cup$ ...

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### Solution of infinite dimension linear system

Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$.
For fix n,
we can construct n dimension linear equation ...

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### Cholesky decomposition of a large covariance matrix

I have a tricky problem concerning a covariance matrix cholesky decomposition.
What I need is to obtain the cholesky decomposition of the estimated variance matrix of the set of samples stored in a ...

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### 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 ...

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### Using Marchenko - Pastur type Theorems on Regression Analysis

Sometimes when doing regression analysis, we estimate our function $g(x) = E(Y |X =x )$ using an orthonormal series, and in particular we use an approximate series $g_{p_n}(x) = \sum_{k=1}^{p_n} ...

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### Stabilization of the pencil of skew symmetric matrices by the orthogonal group

Good morning everybody.
During my researches I've come across the following question.
Let $A$ and $B$ be a couple of square $k\times k$ skew symmetric matrices on $\mathbb R$. Let us consider the ...

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### Any generic way to move a psd matrix to its neighbors?

Given a two positive matrices $A,B$. For simplicity, let's assume that $Tr A=Tr B=1$. Assume that $\|A-B\|_1\leq\varepsilon$, for some small $\varepsilon>0$, where $\|\cdot\|_1$ is the $l_1$-norm, ...

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### How to determine the distance between two matrices under the meaning of a matrix function? [closed]

Suppose a nonlinear infinitely continous differentiable function $f:\mathbb{D}\mapsto \mathbb{R^+}$, where $\mathbb{D}\subset\left\{X|\text{rank}{X}=2,X\in\mathbb{R}^{3\times 3}\right\}$ is a ...

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### Cholesky decomposition of a positive semi-deﬁnite

We know that a positive deﬁnite can be done for Cholesky decomposition,but I want to know that how a positive semi-deﬁnite be done for Cholesky decomposition?The following sentences come from a ...

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### how to find all the solutions to $I+A+\cdots+A^n=0.$ [closed]

Let $GL_3(\mathbb{Z}[i])$ be the group of invertible $3\times 3$ matrices whose coefficients are Gaussian integers.I want to find all the pair $(A\in GL_3(\mathbb{Z}[i]),n\in\mathbb{Z})$ satisfying
...

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### Kernel of a projection

Given $m<n$. Suppose that $H$ and $K$ be $m \times n$ and $n\times (n-m)$ matrices such that rank$(H)=m$, rank$(K)=n-m$, and $HK=0$. For fixed non singular symmetric matrix $A$ define
...

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### Perturbation of Cholesky decomposition for matrix inversion

I am looking for a computationally cheap way to compute $x$ such that $$(L L^T + \mu^2 I)x = y$$
where $L \in \mathbb{R}^{n \times n}$ is a lower triangular definite positive matrix (with some very ...

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### A generalization of van der Waerden's conjecture

I am wondering if the following generalization of van der Waerden's conjecture is true.
Suppose A is an n x n non-negative matrix with all column sums equal to 1, and the sum of row i equal to $T_i$. ...

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### A Problem on Linear Algebra

I'm trying to calculate an integral over the generalized Poincare upper half plane, then I find that I need to show the following identity:
Let $X=(X_{i,j})\in\mathrm{GL}(n,\mathbb R)(n\geq 3)$ ...

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### On a determinant inequality of positive definite matrices

Assume that $B$ and $A$ are two positive definite matrices. Take $B^*$ a block diagonal matrix with block $B_{11}$ and $B_{22}$ of $B$. This means the following:
$$
B=\left[\begin{array}{ll}
...

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### 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 ...

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### Reachability in graphs using adjacent matrix

Assuming a graph $G$ with $N$ nodes distributed in a $\mathcal{L}\times\mathcal{L}$ area randomly. There is an edge between two nodes if and only if the Euler distance between them is equal or less ...

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### Inverse of sparse matrix is not generally sparse [closed]

I have a question regarding inverse of square sparse matrices(or can be restricted to real symmetric positive definite matrices).
I encountered several times the web pages which states that the ...

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### What is the significance of matrix ordered algebras?

I am trying to grok matrix ordered operator algebras, but I am having a hard time understanding their significance from the definition. Here is the definition (or at least, one way of stating it):
...

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### adjacency matrix of random geometric graphs [closed]

Consider a graph with N nodes. All nodes are distributed as a Poisson point process with density of λ in a L*L area. There is an edge between two nodes if and only if the distance between them is less ...

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### Structure theorem for finite dimensional $C^*$-algebras and their representations

I would like a source for some Artin-Wedderburn type facts about these algebras which seem to have easy proofs, and are probably written somewhere.
Let $\mathcal{A} \subset M_n(\mathbb{C})$ be an ...

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### Lower bounds on matrix eigenvalues

Let $A$ be a real $n\times n$ matrix and let $\mu_1, \dots, \mu_n$ the (generalized, complex) eigenvalues of $A$. Assume that
$$ 0 < \alpha < \mathrm{Re}(\mu_1) < \dots < ...

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### Determinants in Graph Theory

In graph theory, we work with adjacency matrices which define the connections between the vertices. These matrices have various properties in themselves. For example, their trace can be calculated (it ...

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### 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 ...

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### Books or references on multidimensional matrix operations [closed]

Have the 2D matrix operations been generalized to n-dimensional matrices?
Are there any books that define various operations on multidimensional matrix? I'd like to see operations such as ...

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### On the linear transformation between matrix space

Suppose we have two matrix subspaces, $n\times n$ matrix subspace $S_1$ and $m\times m$ matrix subspace $S_2$. Every element of $S_1$ and $S_2$ is complex symmetric matrix.
Suppose there exists ...

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### How to understand the matrix behind a Hamiltonian?

thanks to the answers I received to my previous questions, I could derive correctly an elegant partition function for my problem which resembles a second quantized model taking the particles to be ...

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### Possible diagonal values of a product of matrices with some specific characteristics

Hello all,
This is a question that might or might not be related to my previous one.
Imagine you have two matrices:
Matrix $\mathbf{\Phi}=[\Phi_1,\ldots,\Phi_M]\in\mathbb{R}^{L\times M}$ where ...

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### Matrix Inverse with Same Principal Minors

Given an invertible matrix $A \in \mathbb{R}^{n \times n}$, and index set $\langle n\rangle = \{ 1, \dots, n \}$, and the submatrix $A(\alpha)$ with the columns and rows of $A$ with indices $\alpha ...

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### A spectral radius inequality

Define $\rho(A)$ to be the spectral radius of a square matrix $A$. Let $S$ and $T$ be two non-negative square matrices and $h$ a real number such that $\rho(S+T) < h$. Show that $\rho((hI-S)^{-1}T) ...

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### Is Ryser's conjecture on permanent minimizers still open?

Let $A(k,n)$ be the set of $\{0,1\}$ matrices of order $n$ with all their line sums equal to $k$.
Conjecture number 5 on the list from Minc's book, attributed to Ryser, says that if $A(k,n)$ ...

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### Average size of determinants of integer matrices?

I am interested in estimating how large determinants of matrices tend to be 'on average' given the following model: suppose we form $n \times n$ matrices $M$ such that all of the entries of $M$ are ...

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### A short question about the DFT matrix

Is the DFT matrix the unique* unitary matrix with all entries of same magnitude?
(*up to some trivial transformations)

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### Characterizing the singular values of a matrix with structure

Suppose we have a function from $\mathbb{R}^2\to\mathbb{C}$,
$$f(x,y) = e^{\imath\pi x g(y)}$$
where $g(y)$ is periodic in $y\in[-T, T),\ T<\infty$ (e.g., a sinusoid) and $0\leq x < \infty$
...

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### Matrix Transpose Similarility

A famous problem in linear algebra is that
"A $n\times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$."
I know one proof using Smith Normal form. However, I want to know an ...

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### On matrix norms

It is standard to define an induced matrix norm $|||\cdot|||$ from a vector norm $||\cdot||$ in this way:
$|||A|||=\max_{x \neq 0}{\frac{||Ax||}{||x||}}$.
Suppose we define a different function of ...

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### Ring-theoretic version of a matrix problem

Problem #17 in Zhan's survey of open problems in matrix theory is the Li-Poon problem on writing a square real matrix as the linear combination of $k$ orthogonal matrices. They proved that it is ...

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### system of homogeneous matrix equations

Edit: Maybe I should make it clear that by a solution I mean a pair of matrices $(A,B)$ such that the identity below holds for any complex numbers $x,y$.
One of my friend asked me the following ...

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### Matrix groups and presentation

Suppose $K$ is a number field and I have a subgroup of $GL_2(K)$ for which I know a (finite) set of generators. Is there an algorithm that gives me a presentation of the group?
More precisely, the ...

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### 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 ...