# 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 ...

2,044 questions

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### Simple way to construct e (or characterize) real $n \times n$ matrices $K$ satisfying $(-1)^{|A|}\det(K-I_A) \ge 0,\;\forall A \subseteq [\![n]\!]$

Let $N$ be a large positive integer.
Question
What is a simple way to construct (or characterize) real $N \times N$ matrices $K$ such that
$$
s_A(K):=(-1)^{|A|}\det(K-I_A) \ge 0,\;\forall A \...

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### Matrix Product Chain Representation of an Addition Chain

I am looking for references to anything interesting thats know about matrix product chains that take the vector $\{1\}$ to another vector $\{n\}$ (the end result of an addition chain). Each matrix ...

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### Matrix calculation of collision in 3D-plane [on hold]

Given a sphere(S) with the center a(2,0,0) with the radius of √2. A spherical object flying with the direction R=(-1,-2,0) hits (S) at point t=(3,1,0) and is reflected on it. Compute the direction of ...

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### On the existence of fixed points of a matrix iteration

Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e. all eigenvalues of $A$ have negative real part). Let $\succeq$ denote the standard partial order in the cone of positive semidefinite ...

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### A question related to (random) matrix factorization

Let $\mathbf{B}$ be an $m\times r$ binary matrix taking values in $\{0,1\}$, and assume that $m\geq r$. I am wondering what can be said about the recovery of $\mathbf{B}$ from $\mathbf{B}\mathbf{B}^T$....

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### A $\mathbb C$-linear map from $M(p-1,\mathbb C)$ to $\mathbb C^\hat G$, where $p$ is an odd prime and $G=\mathbb Z/(p) ^\times$

Let $p$ be an odd prime and $G=(\mathbb Z/(p))^\times=\{1,2,...,p-1\}$ i.e. $G$ is a cyclic group of order $p-1$. Let $\hat G:=\{\chi:G \to \mathbb C^\times : \chi $ is a group homomorphism $\}$. For ...

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### Hadamard matrix research [closed]

I need to find out whether something new has happened in the study of Hadamard matrices for the last 10 years.
If someone was engaged in matrices, can you recommend literature?
K. Horadam is the ...

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### Existence of orthogonal basis of symmetric $n\times n$ matrices, where each matrix is unitary?

For a positive integer $n$, let $S_n$ denote the set of $n\times n$ symmetric matrices over $\mathbb{C}$. As a complex vector space, this set has dimension $\mathrm{dim}(S_n)=\binom{n+1}{2}$. The ...

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### k-rank numerical range of an operator

Let $T\in\mathscr{B(\mathcal{H})}$ where $\mathcal{H}$ is an infinite dimensional seperable Hilbert Space and $k\in\mathbb{N}\cup\{\infty\}$. Now we define k-rank numerical range of $T$ denoted by $\...

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### Largest eigenvalue of product of orthogonal-projection rank-1 perturbation

Suppose I have a symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ with $n$ linearly indepedent columns $a_1,...a_n$ in $\mathbb{R}^n$. All columns $a_i$ has norm 1, but they are not ...

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### Set of integer matrices $A$ such that $(A^k)_{k\in\mathbb{N}}$ is eventually periodic

This is a more elegant version of the original version which can be found below; it is based on a suggestion by Peter Mueller.
Let $\mathbb{N}$ denote the set of positive integers and for $n\in\...

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223 views

### How to verify the characteristic polynomial? [closed]

I am computing the characteristic polynomial of a matrix over a number field, using the minimal polynomial of it. Is there a fast way to verify the characteristic polynomial of a big matrix ?

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### A generalization of matrix minors to non-integer values

I am interested to know if there exist a notion of $k$-minors of a real square matrix, for non-integer positive values of $k$
One approach I thought of was to use the fact that the $k$-minors are (...

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### Computing with a vector subspace equipped with a prescribed basis over finite fields

Let $U$ be a subspace of the finite dimensional vector space $V$ over a field $\mathbb{k}$. Let $B_V$ and $B_U$ be fixed bases for $V$ and $U$ respectively. Let $u \in U$ and let's give ourselves $[u]...

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### matching two positive-semidefinite matrices

Let $M_1$ and $M_2$ be two real positive-semidefinite matrices. Is there any algorithm to compute a permutation matrix $P$ to minimize $\| M_1-PM_2P^T \|_F^2$ or equivalently to maximize $trace(...