Related papers: Eigenvector of a matrix in $SO_3(\mathbb{R})$

Eigenvectors of tensors, as studied recently in numerical multilinear algebra, correspond to fixed points of self-maps of a projective space. We determine the number of eigenvectors and eigenvalues of a generic tensor, and we show that the…

A symmetric matrix $M=(m_{ij}) \in \mathbb{R}^{n \times n}$ is said to be associated with an $n$-vertex graph $G=(V,E)$ with vertex set $\{v_1,\ldots,v_n\}$ if, for every $i \neq j$, we have $m_{ij} \neq 0$ if and only if $\{v_i,v_j\}\in…

A simple approximate relationship between the ground-state eigenvector and the sum of matrix elements in each row has been established for real symmetric matrices with non-positive off-diagonal elements. Specifically, the $i$-th components…

Let $x_1, \dots, x_n$ be points in a metric space and define the distance matrix $D \in \mathbb{R}^{n \times n}$ by ${D}_{ij} = d(x_i, x_j)$. The Perron-Frobenius Theorem implies that there is an eigenvector $v \in \mathbb{R}^n_{}$ with…

Consider $n$ linearly independent vectors in $\mathbb{C}^n$ which form columns of a matrix $A$. The recursive evaluation of eigen directions (normalized eigenvectors) of $A$ is the solution of an eigenvalue problem of the form…

Eigenvector continuation is a computational method that finds the extremal eigenvalues and eigenvectors of a Hamiltonian matrix with one or more control parameters. It does this by projection onto a subspace of eigenvectors corresponding to…

Let $m,n>1$ be integers and $\mathbb{P}_{n,m}$ be the point set of the projective $(n-1)$-space (defined by [2]) over the ring $\mathbb{Z}_m$of integers modulo $m$. Let $A_{n,m}=(a_{uv})$ be the matrix with rows and columns being labeled by…

In this paper, we define a matrix which we call Vieta matrix and calculate its determinant: $$ \left( \begin{array}{cccc} 1&1&\cdots&1\\ a_{2}+a_{3}+\cdots+a_{n}&a_{1}+a_{3}+\cdots+a_{n}&\cdots&a_{1}+a_{2}+\cdots+a_{n-1}\\…

The method of computing eigenvectors from eigenvalues of submatrices can be shown as equivalent to a method of computing the constraint which achieves specified stationary values of a quadratic optimization. Similarly, we show computation…

The singular values $\sigma >1$ of an $n \times n$ involutory matrix $A$ appear in pairs $(\sigma, \frac{1}{\sigma}),$ while the singular values $\sigma = 1$ may appear in pairs $(1,1)$ or by themselves. The left and right singular vectors…

In this paper we are concerned to find the eigenvalues and eigenvectors of a real symetric matrix by applying a new numerical method similar to Jacobi method. Our approch consists to use a new orthogonal matrix. The computation of the…

In this paper we study the structure and give bounds for the eigenvalues of the $n\times n$ matrix, which $ij$ entry is $(i,j)^\alpha[i,j]^\beta$, where $\alpha,\beta\in\Rset$, $(i,j)$ is the greatest common divisor of $i$ and $j$ and…

In this paper we bring to light an unprecedented property of the eigenvalues of a matrix A with the eigenvalues and eigenvectors of a submatrix of A. This property can be used, through the technique developed here, to determine some of…

We characterize all graphs for which there are eigenvectors of the graph Laplacian having all their components in {-1,+1} or {-1,0,+ 1}. Graphs having eigenvectors with components in {-1,+1} are called bivalent and are shown to be the…

In this article, we prove the finiteness of the number of eigenvalues for a class of Schr\"odinger operators $H = -\Delta + V(x)$ with a complex-valued potential $V(x)$ on $\bR^n$, $n \ge 2$. If $\Im V$ is sufficiently small, $\Im V \le 0$…

We investigate self-adjoint matrices $A\in\mathbb{R}^{n,n}$ with respect to their equivariance properties. We show in particular that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group…

Let $S$ be an orthogonal array $OA(d,k)$ and let $c$ be an $r$--coloring of its ground set $X$. We give a combinatorial identity which relates the number of vectors in $S$ with given color patterns under $c$ with the cardinalities of the…

In 2011, Haemers asked the following question: If $S$ is the Seidel matrix of a graph of order $n$ and $S$ is singular, does there exist an eigenvector of $S$ corresponding to $0$ which has only $\pm 1$ elements? In this paper, we construct…

Given a $(k+1)$-tuple $A, B_1,...,B_k$ of $(m\times n)$-matrices with $m\le n$ we call the set of all $k$-tuples of complex numbers $\{\la_1,...,\la_k\}$ such that the linear combination $A+\la_1B_1+\la_2B_2+...+\la_kB_k$ has rank smaller…

Given two positive integers $n$ and $k$ and a parameter $t\in (0,1)$, we choose at random a vector subspace $V_{n}\subset \mathbb{C}^{k}\otimes\mathbb{C}^{n}$ of dimension $N\sim tnk$. We show that the set of $k$-tuples of singular values…