# 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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### Does there exist an axiomatic unsupervised approach for Link prediction based on either distances or matrices?

Does there exist an axiomatic unsupervised approach for Link prediction based on either distances (in my pasted link - related to the Graph theory's Closeness centrality) or matrices (i.e., if we fake ...
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### Let A be a 4 × 4 matrix. Let λ be an eigenvalue of A. Suppose the geometric multiplicity of λ is 2 [closed]

Prove that if A is diagonalizable, then there exists a0, a1, a2 ∈ R such that A^3 + a2A^2 + a1A + a0I4 = 0. I don't even know where to start. I've tried implementing a method similar to this, but I ...
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### Low-rank factorization of a Finite Element matrix

I have a matrix $M\in \mathbb R^{n\times n}$, stemming from a Finite Element discretization of an advection function. I want to find a factorization $M= S E T$ with $S, T\in \mathbb R^{n\times s}$ ...
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### Find good representatives for something related to the orbit of $\mathrm{GL}_n\times \mathrm{GL}_m$ acts on $M_{n,m}$

$\DeclareMathOperator\GL{GL}$Fix a field $F$, and we only talk about spaces and groups over it. We consider the $n\times m$ matrix space denoted by $M$. And $\GL_n \times \GL_m$ acts canonically on $M$...
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1 vote
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### Non-vanishing principal minors up to swapping columns

An undergraduate student asked me the following seemingly easy question. After a few days of thinking, I still couldn't come up with an answer, nor could I find one online. Maybe folks here could help?...
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### Are these two methods for constructing Hadamard matrices known?

These two observations came while researching the empty set of odd perfect numbers and unitary perfect numbers: Context: Let $n$ be a natural number and $D_n$ be the set of divisors. We can make this ...
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### An determinant inequality of strictly diagonally dominant matrix [migrated]

Recently, I have seen a matrix inequality but don't know how to prove it. The inequality goes as follows. Matrix A is a strictly diagonally dominant matrix. image of inequality I encounter it in my ...
1 vote
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### Pre-positive definite functions?

A function $f(x,y)$ is positive definite if matrices $( f(x_i, x_j) )_{i, j \in F}$ are positive definite for all finite index sets $F$. This is frequently hard or impossible to check given some ...
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### Existence of cyclic subspace decompositions for pairs of commuting matrices

Let $\mathbb{K}$ be an arbitrary field (possibly finite). Let $V$ be a finite-dimensional vector space over $\mathbb{K}$, and let $A,B$ be two linear endomorphisms of $V$ which commute. For $v\in V$, ...
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### Strassen factorization for $n \geq 3$ [migrated]

There is the famous Strassen matrix multiplication method (see here for further information). In essence it boils down to the fact, that we can multiply 2x2 matrices with each other and don't have to ...
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### Interpretate a condition on Signed graph

In my research I came up with a condition on signed complete graph as follows: Consider adjacency matrix of signed complete graph, i.e. symmetric, diagonal-free matrix with elements $A_{ij}=-1$ or $+1$...
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### Inductive Cholesky decomposition to prove that a function is positive definite over the natural numbers?

I am trying to prove that the function: $$k(a,b):=\frac{1}{\operatorname{rad}\left ( \frac{ab}{\gcd(a,b)^2} \right )}$$ is a positive definite function over the natural numbers. What has sometimes ...
507 views

### Jensen-like inequality for random matrix: $\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$

Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution. What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is ...
1 vote
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1 vote
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### Matrix valued word embeddings for natural language processing

In natural language processing, an area of machine learning, one would like to represent words as objects that can easily be understood and manipulated using machine learning. A word embedding is a ...
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### What is the quantity $\sqrt{\frac{c^2+d^2}{a^2+b^2}}$ of a matrix with determinant one?

Suppose that $A \in \mathbb R^{2 \times 2}$ has determinant one, $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ I'm currently working on a problem where I obtained a condition on the ...
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### An inequality related to matrix trace

$$Tr(A|A^T U \Sigma U^T|) \leq Tr(AA^T U \Sigma U^T)$$ where $A \in \mathbb{R}^{m\times n}$ is a real rectangular matrix, $U \in \mathbb{R}^{n \times n}$ is a orthonormal matrix and $\Sigma$ is a ...
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### Conditions for distinct nonzero eigenvalues in product DAD for symmetric matrix A with repeated nonzero eigenvalues and diagonal matrix D

Let $A\in\mathbb{R}^{n\times n}$ be an real symmetric matrix with eigenvalues $\lambda_1, \lambda_2, \cdots, \lambda_n$ with some of which be nonzero and repeated, i.e., there exist $\lambda_i \ne 0$ ...
1 vote
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### square matrix depending on complex value: spectral radius continous? [closed]

Let $A(z)$ be a $n\times n$ square matrix depending on the complex value $z$ and $\lambda_z$ is its spectral radius. Is $\lambda_z$ continous or is it possible that it can jump? Or maybe someone knows ...
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### Norm of matrix product sum

Given matrices $A_{n\times n}, B_{n\times m}, C_{m\times m}$ such that $A^iBC^{N-i}$ is matrix with all zeros except upper right element for all $i$ from $0$ to $N$, what can we say about Frobenius ...
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### Diameter of the unimodular group with Gauss moves

$\DeclareMathOperator\GL{GL}$Consider the unimodular group $\GL_n(\mathbb{Z})$, consisting of integral matrices $A \in \mathbb{Z}^{n \times n}$ such that that $\det(A) =\pm 1$. It is well known that ...
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1 vote
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### Optimal transport between two matrices

I'm investigating the use of optimal transport to define a distance between non-negative matrices that satisfy the condition $\mathbf e^\top\mathbf M\mathbf e = 1$ (i.e., the sum of all elements in ...
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### Are there any known lower complexity bounds on solving positive semidefinite or positive semidefinite feasibility problems?

I've been trying to attack the problem posted here, about quickly checking if a matrix has any positive semidefinite completions. I suspect that the answer to the question is "no", because ...
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### Is there a name for this family of matrices?

Let $0<a_1<a_2<\cdots<a_n$ and let $A$ be the symmetric $n\times n$ matrix with ${ij}^\text{th}$ entry $A_{ij}=\min\{a_i,a_j\}$. For example, if $a_i=i$ for each $i\le n=5$ then A=\begin{...
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### How to represent infinite matrices in Mathematica?

I asked this question on Mathematica site and the answer was basically "it is impossible". So, some math theory needs to be thrown in besides coding. Let us represent infinite matrices as ...
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### What conclusions can I derive from this family of trace inequalities?

Problem. Let $n_1,\ldots,n_s,m_1,\ldots,m_s\ge 0$ be nonnegative integers and set $m := \sum_{i=1}^s m_i$ and $n := \sum_{i=1}^s n_i$. Let $\oplus$ be an operation on matrices which stacks them in a ...
1 vote
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### Matrix sparsity pattern from Boolean condition

I would like to determine the sparsity pattern of a matrix $A_m$, where the entry $(k,l)$ of $A_m$ is non-zero if the condition $\operatorname{test}(k,l,m)$ is true: $\operatorname{test}(k,l,m)$ = &...
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### Bound on the number of connected components of a linear algebraic group $G<\mathrm{SL}_n$?

Let $G<\mathrm{SL}_n$ be a linear algebraic group defined over a field. Is there a bound on the number of connected components of $G$ in terms of $n$ alone? (The bound will evidently not be any ...
1 vote
I recently need to apply the following concept of product of two 2d tensors to create a 3d tensor (tensors understood as generalized arrays): given two 2d tensors $A_{m\times n}$ and $B_{n\times p}$, ...
What is the minimal number of linearly dependent rank-1 projectors $\vec v \vec v^t$ in dimension n, under the condition that every set of n column vectors $\vec v$ is linearly independent. PS: the ...