# Questions tagged [eigenvector]

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### Given the eigenvalues of a matrix, can one find the eigenvectors? [closed]

If only the eigenvalues and the dimensions of a square matrix are given, is it possible to find the eigenvectors or more information about the matrix? I am trying to find $SS^{-1}$, but I'm a bit ...
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### Eigenvalues of a rank-one update of a symmetric matrix

I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda_i$, then I add another matrix to it as: $A+xx^\top$ where $x$ $(n\times 1)$ is a column vector. and also $A=yy^\top$ with $y$ a $(n-1)$ rank ...
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Let $A=(a_{kl})$ be a matrix in $M_n(\mathbb{R})$ when $n$ is even. Let $B=(b_{kl})$ be the symmetric $2n$ by $2n$ matrix whose entries are given by, $b_{k,l}=a_{kl}$ if $1\leq k,l\leq n$. $b_{n+k,l}=... 3 votes 0 answers 41 views ### What is the supremum of 1-dim Hausdorff measure of the nodal set of Neumann eigenfunction$u$for planar convex domain All descriptions of this question are limited to 2-dimension for simplicity. Recently, I read some papers on the nodal set of Laplacian eigenfunctions. Denote$\Sigma=\{u(x)=0\}$be the nodal set of a ... 3 votes 2 answers 238 views ### Eigenvalues and eigenvectors of non-symmetrical tridiagonal matrix The question is the following: given a matrix $$A=\begin{pmatrix} 1& 2 & & & & \\ 1& 0& 1 & & & \\ & 1& 0& 1 & &\\ & &... 0 votes 0 answers 76 views ### Diagonalizing a specific case of symmetric block matrix Let's consider the following block matrix$$ M = \begin{pmatrix}D&A^T\\A&-D\end{pmatrix},$$where A and D are n \times n matrices. The diagonal matrix D is defined by D_{kk} = k \... 0 votes 1 answer 102 views ### Are there zero entries in the eigenvector corresponding to a simple eigenvalue? For a real symmetric matrix M and a simple eigenvalue \lambda, under which conditions the corresponding eigenvector has no zero entries? Perhaps, this is unconditional and one can provide a proof? 4 votes 1 answer 157 views ### Least squares problem with left and right unknowns For i=1,...,n, let b_i be a scalar and A_i be an k\times l matrix. Is there a closed form solution for the following problem assuming n>k+l?$$\min_{x\in \mathbb{R}^k ,y\in \mathbb{R}^l} \... 0 votes 0 answers 28 views ### upper band for the eigenvalue-distance from an eigenvalue regarding its eigenvector extension, in the adjacency matrix of a local graph? Background: Suppose we have a system in which an electron can hop locally on this lattice. Here locally means that the electron can hop up to a short distance. We can make an energy spectrum that ... 10 votes 2 answers 420 views ### Support of eigenvectors Consider the$N$by$N$matrix $$M_N= \begin{pmatrix} 1+3\lambda & -1-2\lambda & - \lambda & 0 & 0 &0 &0\\ -1-2\lambda & 2+3\lambda & -1 & -\lambda & 0 & 0 &... 4 votes 0 answers 63 views ### Local energy estimate in a semiclassical regime Let us consider h_n=(2n+1)^{-1/2}\to 0 as n\to \infty be a small parameter, which we just write as h for convenience, and u_h : \mathbb{R} \to \mathbb{R} be functions satisfying Pu_h=0 (I ... 2 votes 1 answer 116 views ### On the eigen vectors of a diagonalizable matrix Let us consider the space M_n(\mathbb{C}). By a unitary matrix U=(u_{ij}) we mean that U^{-1}=(\overline{u_{ji}}). Q. Let U be a unitary matrix. I am looking for the pairs of matrices (D,A) ... 1 vote 0 answers 156 views ### Find the eigenvectors from the QR algorithm in the unsymmetric case It is possible to find many references describing the QR Algorithm with more or less refinements to approximate the eigenvalues of a square matrix A\in\mathbb{R}^{n\times n}. I implemented a version ... 1 vote 1 answer 413 views ### Diagonalizing a symmetric block matrix Let us consider the matrix$$ A = \begin{pmatrix} a & c+ib \\ c-ib& a \end{pmatrix},$$then this matrix has eigenvalues a\pm \sqrt{c^2+b^2}. Now, let us consider a block matrix$$ A = \begin{... 5 votes 0 answers 198 views ### Perturbation of Neumann Laplacian Consider the$N \times N$matrix $$A_{\alpha}=\begin{pmatrix} \lambda_1 & -1 & -\alpha & 0 & \cdots & 0\\ -1 & \lambda_2 & -1 & -\alpha & \cdots & 0\\ -\alpha &... 4 votes 2 answers 316 views ### How close to uniform are Perron-Frobenius eigenvectors? Let A=(a_{i,j}) be a square matrix with non-negative entries. (Assume A is symmetric, if it helps.) Let v be a Perron-Frobenius eigenvector. What do we need to assume about A in order to have ... 1 vote 0 answers 309 views ### When must an eigenvector have only non-negative entries? What would be a reasonable sufficient condition on a real symmetric matrix that would force its eigenvector with largest eigenvalue (or one of its eigenvectors with maximal eigenvalue) to have only ... 1 vote 0 answers 110 views ### Eigenvalues and eigenvectors of non-symmetric elliptic operators We know that the operator A=\Delta with domain D(A)=\{u\in W^{2, 2}(\Omega): u=0 \ \ \text{on } \partial\Omega\} (say \Omega is a bounded nice domain) has eigenvalues \lambda_1>\lambda_2\ge \... 1 vote 1 answer 263 views ### Trace minimization for generalized eigenvalue problem In , it is shown in theorem 1.2 that for symmetric n \times n matrices A, B, we have$$ \min_{Y \in Y^*} \text{tr}(Y^TAY) = \text{tr}(X^TAX) = \sum_{i=1}^p \lambda_i, $$with$$ \text{$X^...
I was wondering what we can say about the eigenvectors of a matrix $A$ fullfilling $Ax =0$ where $A$ is symmetric with a diagonal equal to one and every row sums up to 0. Obviously this is a ...