# Tagged Questions

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### Equivalence of Positive Matrix in Infinite Dimensional Vector Space

What is the corresponding linear operator on an infinite dimensional vector space, say a Banach space or Hilbert space, to the nonnegative matrix on a finite dimensional vector space? What is the ...
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### Existence of a real eigenvalue

I have a matrix $M \in \mathbb{R}^{(n+1) \times (n+1)}$ that is tridiagonal. In numerical computations I found out that I always find a real eigenvalue. My question is: Is there a theorem that ...
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Let $G$ be an undirected, simple, bipartite graph with parts $V$ (having $n$ vertices) and $W$ (having $m$ vertices). Define the following $n$-by-$n$ matrix: for any $i,j \in V$, $$a_{ij} = |N_i \cap ... 0answers 176 views ### An optimization problem on the sphere Let S be a sphere centered at origin in \Bbb R^{2n} of radius \sqrt{2n}. Let D be a diagonal matrix. Let U be an orthogonal matrix. Let r\in\Bbb Z_+ be a fixed integer. Let vector ... 0answers 76 views ### “Almost orthogonalizing” matrices using a signature matrix Suppose A and B are two real symmetric n \times n matrices (If simpler, consider A and B to be 0/1 matrices, say, adjacency matrices of d-regular graphs). Then ||AB||_{op} \leq ... 1answer 209 views ### Separating the spectrum of a Hermitian matrix Given Hermitian matrix A, I would like to perturbate it so that its eigenvalues become well-separated. Specifically, let A be some Hermitian matrix, and let G be a Gaussian matrix, with each ... 1answer 172 views ### positive semidefinite matrix condition There is a great work of Alizadeh that in section 4 speaks about Minimizing sum of the first few(k-largest) eigenvalues of a symmetric matrix. Instead of a symmetric model we use the weighted ... 0answers 96 views ### Perturbation of spectrum and eigenspaces Let A \in \mathbb{C}^{n \times n} be an n \times n matrix. Consider the rank-1 perturbation A' of A given by replacing a column v of A by \alpha \cdot v, where \alpha \in [0, 1). Can ... 1answer 307 views ### numerical range of a column-zero-sum matrix I am trying to produce an example of a (necessarily non-normal) matrix that has only eigenvalues with positive real part, but whose numerical range contains elements with strictly negative real part. ... 1answer 377 views ### square root of a certain matrix [closed] Hello, I'd like to know the square root of the following n by n matrix, for n > 2 and r>0: R_{ii}=r+1 for i < n R_{ij}=r otherwise The 2 by 2 case is given by ... 0answers 65 views ### Second eigenvalue of a weighted tree Hello, I am interested in upper bounding the second largest eigenvalue of the adjacency matrix of a graph T with the following property: 1. T contains self loops. 2. T contains multiple edges ... 1answer 327 views ### Eigenvalues of Sum of non-singular matrix and diagonal matrix Suppose D={\rm diag}(d_i) is a diagonal matrix with all diagonal entries d_i=\pm 1. This implies D^2=I. Suppose A is a non-singular Hermitian matrix. If we know that A+A^{-1}+D has rational ... 0answers 111 views ### Optimization over Spectral Laplacian in cycles and trees Is there any idea on how one can deal with an optimization problem of sum of k largest eigenvalues(min) of Laplacian matrix of a simple cycle or tree? I would like to use semidefinite programming for ... 0answers 48 views ### 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 ... 1answer 193 views ### 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) ... 2answers 313 views ### spectral radius monotonicity I encountered an inequality when reading a paper. Can someone help to show how to prove it? Let be the spectral radius of matrix A or \rho(A)=\max\{|\lambda|, \lambda \text{ are eigenvalues of ... 0answers 88 views ### Global solution for spectral clustering I used spectral clustering for directed graphs suggested by Dengyong Zhou paper to partition the graph.I selected the eigen vectors corresponding to k largest eigen values and then I use kmeans or FCM ... 2answers 360 views ### Eigenvalues of principle minors Vs. eigenvalues of the matrix Say I have a positive semi-definite matrix with least positive eigenvalue x. Are there always principal minors of this matrix with eigenvalue less than x? (Here "semidefinite" can not be taken to ... 10answers 3k views ### real symmetric matrix has real eigenvalues - elementary proof Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers? 1answer 338 views ### relationship between eigenvalues of (A-B) and eigenvalues of (A^2-B^2) Let us suppose that A_{n} and B_n are sequences of positive definite matrices satisfying c\leq \lambda_{\min}(A_n)\leq \lambda_{\max}(A_n)\leq C and c\leq \lambda_{\min}(B_n)\leq ... 1answer 247 views ### A is a nonnegative matrix; the only principal submatrix having spectral radius above 1 is A itself Let \rho(M) denote the spectral radius (modulus of the largest eigenvalue) of a square matrix M. I am looking for a characterization or anything else interesting about the set of matrices A ... 1answer 156 views ### Effects of unitarian multiplication into the spectrum of a finite matrix. I am interested in the following problem: Let P be a n\times n complex finite matrix such as PP^\dagger =W. Given W, what can I say about the spectrum of P? This matrix "square-root" has ... 1answer 109 views ### Morse index and permutation of diagonal entries of a symmetric matrix Do there exist results concerning preservation or not of the Morse index of a symmetric matrix A, after permuting its diagonal entries, and keeping fixed the off--diagonal ones? Thanks! 3answers 421 views ### Has the largest-to-rest eigenvalue ratio of real symmetric matrices been researched before? I'm investigating the eigenvalue ratios$$ \frac{\lambda_1}{\sum_{j=2}^N\lambda_j} \quad\mbox{and}\quad \frac{\sum_{j=1}^N\lambda_j}{\sum_{j=2}^N\lambda_j} $$of the NxN matrix B=AA^T. \lambda_1 ... 0answers 228 views ### Eigen-decomposition perturbation Let A, B and A_k + B be symmetric matrices with eigenvalues \sigma_1 \geq \sigma_2 \ldots \geq \sigma_n, \rho_1 \geq \rho_2 \ldots \geq \rho_n and \lambda_1 \geq \lambda_2 \ldots \geq ... 2answers 937 views ### Minimum off-diagonal elements of a matrix with fixed eigenvalues Hello, I am en engineer working in radar research. I came accross a problem I cannot seem to find math literature on it. I can ask it in two different ways. Perhaps depending on the reader, the ... 1answer 146 views ### Generalizing the spectral radius of a unistochastic matrix Consider a square matrix A, and from it construct B whose entries are the squared magnitudes of those in A. What can we say about the spectral radius of B? I know that for a unitary matrix ... 2answers 2k views ### Interesting relationships between Cholesky decomposition and diagonalization Let \Sigma be a hermitian positive definite matrix and L be it's Cholesky decomposition so that LL^\ast=\Sigma. Furthermore, let's diagonalize \Sigma as \Sigma = P\Lambda P^\ast. \Lambda ... 0answers 451 views ### Bounding sum of first singular values squared for Kronecker sum of traceless matrices Let A and B be 4\times4 traceless matrices with Hilbert-Schmidt norms summing up to 1/4, i.e.$$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + ...
Let $T\in \mathbb{R}^{n\times n}$ be a symmetric tridiagonal matrix having the off--diagonal entries equal to -1. The diagonal entries are all positive, $a_i>0$, $i=\overline{1,n}$, and there ...