# Questions tagged [eigenvalues]

eigenvalues of matrices or operators

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### Subleading terms in Weyl's Law

The two term Weyl's conjecture states that $$N(\lambda)\sim\frac{\operatorname{area}(\Omega)}{4\pi}\lambda-\frac{\operatorname{perimeter}(\partial\Omega)}{4\pi}\sqrt\lambda$$ where $\Omega$ is a ...
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### Eigenvalues of random matrices are measurable functions

I have read that if a random matrix is hermitian then its eigenvalues are continuous, hence also measurable. If the random matrix is not hermitian, the eigenvalues are not continuous in some cases. ...
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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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### When is the sum of matrices (circulant + [super upper triangular]) not diagonalizable?

By the circulant matrix $C \in M_n(\mathbb{R})$, we mean that $$C = \left[\begin{array}{c|c|c|c} e_n & e_1 & \cdots & e_{n-1} \end{array}\right]$$ where $e_1,\dots,e_n$ are the standard ...
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### Prove that sum of eigenvalues of the inverse of an nxn correlation matrix A is greater than or equal to n

I stuck on this question and here is my thoughts: So we have a nxn correlation matrix A with eigenvalues: λ_1,λ_2,...,λ_n 1.According to the property of correlation matrix, (λ_1)+(λ_2) + ... + (λ_n) = ...
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### Generalized eigenvalues of block matrix

Let $A, D \in \mathbb{R}^{n\times n}$ be symmetric matrices and consider the following matrix pencil $$\begin{pmatrix} -I & A+\lambda I \\ A+\lambda I & -D \\ \end{pmatrix}$$ If we already ...
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### Faulty algorithm for simultaneous diagonalization?

I found a simple algorithm for simultaneous diagonalization of two commuting matrices (Nordgren - Simultaneous Diagonalization and SVD of Commuting Matrices), which seemed to be well-founded. For ...
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If $A$ is a Hermitian matrix and $A_j$ the principal minor with the $j$ row and column deleted and $\phi_A(x)$ the characteristic polynomial. The Cauchy interlacing iheorem states that the roots of $\... 2 votes 1 answer 246 views ### Eigenvalues of a specific matrix I have a block matrix $$M=\begin{bmatrix} I_0& I_1& \cdots& I_1\\ I_2& I_0& \ddots& \vdots\\ \vdots& \ddots& \... 0 votes 0 answers 93 views ### The eigenstructure of the symmetric tridiagonal matrix whose entries are a_{kk}=\cos\frac{k\pi}{n+1} and$$a_{1,2}=\cdots=a_{n-1,n}=1$$Suppose that A=(a_{kl})_{k,l=1}^n is a symmetric tri-diagonal matix in M_n(\mathbb{R}) whose diagonal entries are a_{kk}=\cos\frac{k\pi}{n+1} and$$a_{1,2}=\cdots=a_{n-1,n}=1$$Any approach to ... 0 votes 0 answers 60 views ### Eigenvectors of the symmetric tridiagonal matrices whose entries above the diagonal are all the same Let us consider the real symmetric tridiagonal matrix T=(t_{kl}) in M_n(\mathbb{R}) with$$t_{1,2}=t_{2,3}=\cdots=t_{n-1,n}=1$$How can we compute the eigenvectors of T? 1 vote 0 answers 76 views ### Generalized matrix determinant lemma for pseudo-determinant of symmetric matrix The pseudo-determinant of a square matrix A is the product of its nonzero eigenvalues. Consider the generalized matrix determinant lemma$$\det(A+UWV^\top) = \det A\det W\det(W^{-1} + V^\top A^{-1}U)... 2 votes 1 answer 66 views ### The eigenvalues of the product$WD$for some particular matrices Let$D$be a diagonal matrix in$M_{2n}(\mathbb{R})$such that$D^2=I$and Trace$(D)$=0 Suppose that$e_k$s are the standard vectors in$\mathbb{R}^{2n}$, that is $$e_k=(0,\cdots 0,1,0,\cdots,0)^t$$... 1 vote 0 answers 44 views ### What do you call this class of matrices with a unique positive eigenvalue associated to a graph? I am looking for the name of a class of symmetric matrices$M\in\Bbb R^{n\times n}$that I can associate to a (finite simple) graph$G=(V,E)$with$V=\{1,...,n\}$and that have the following ... 3 votes 0 answers 107 views ### What is the computational complexity of Arnoldi algorithm for diagonalization? What is the space and time computational complexity of finding$k$eigenvalues of an$N\times N$matrix using the iterative Arnoldi algorithm? I know that exact diagonalization scales like$O(N^3)$, ... 0 votes 1 answer 69 views ### The spectrum of the product$JA$where$J=I_n\oplus (-I_n)$Let$A$be a real symmetric matrix in$M_{2n}(\mathbb{R})$with$A^2=I_{2n}$. Suppose that the Schur decomposition of$A$is given by$A=\Lambda^t D \Lambda$. Let us consider the following matrix. $$... 4 votes 1 answer 157 views ### largest eigenvalue of the difference between two quadratic forms Let U,V\in\mathbb{R}^{4\times n} such that UU^T=VV^T=I, and A\in\mathbb{R}^{n\times n} be an Hermitian matrix. Is it true that$$\sqrt{\lambda_{\text{max}}\left(\left(UAU^T-VAV^T\right)^2\right)}... 2 votes 2 answers 91 views ### Eigenvalues and eigenvectors of k-blocks matrix I'm trying to find the eigenvalues and eigenvectors of the following$n\times n$matrix, with$k$blocks. \begin{gather*} X = \left( \begin{array}{cc} A & B & \cdots & \\ B & A & ... 1 vote 1 answer 75 views ### Eigenvalues of a circulant: DFT or Inverse DFT Convention? Currently, most engineering texts (and webpages including Wikipedia) define forward discrete Fourier transform with a negative sign on the exponential. This is a convention and the inverse discrete ... 0 votes 0 answers 51 views ### Properties on the eigenvalues of a random binary matrix I'm considering a problem related to the spectral properties (singular values, eigenvalues and eigenvectors) of large, random binary matrices. In case this can help situate my background, I am aware ... -2 votes 1 answer 182 views ### Property of positive semi-definite Let$A$is a positive semi-definite matrix like this: $$A = \begin{bmatrix} 1 & \alpha_{1,2} & \alpha_{1,3} & \alpha_{1,4}\\ \alpha_{1,2} & 1 & \alpha_{2,3} & \alpha_{2,4}\\ \... 4 votes 0 answers 129 views ### Upper bound for the first eigenvalue of the Laplacian on surfaces with boundary Let \Sigma be a compact smooth surface with boundary. Define$$\Lambda(\Sigma) := \sup \{ \lambda_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ is a smooth Riemannian metric on$\Sigma$} \}$$... 3 votes 1 answer 115 views ### On the bounds of the sum of the squares of spectral variation of two real symmetric matrices Suppose A and B are two symmetric real \{0,1,-1\} matrices of order n with diagonal elements as zeros (therefore the traces are zeros) and eigenvalues \lambda_1\ge \lambda_2\ge \dotsb \ge \... 7 votes 1 answer 468 views ### Eigenvalues of the Laplacian on surfaces with boundary Let \Sigma be a compact smooth surface with boundary. Is it true that the supremum$$\sup \{ \lambda_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ smooth Riemannian metric on$\Sigma$} \}$$... 4 votes 1 answer 141 views ### Spectrum near zero of -\partial^2_x + V : L^2(\mathbb{R}) \to L^2(\mathbb{R}), where V = O(|x|^{-2 - \delta}) Let H = -\partial^2_x + V(x) : L^2(\mathbb{R}) \to L^2(\mathbb{R}) be a one dimensional Schrödinger operator, where the potential V is real-valued, belongs to L^\infty(\mathbb{R}), and, as |x| \... 1 vote 0 answers 85 views ### What are the eigenvalues/eigenvectors of the matrix A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^{\frac{n-1}{2}} when n is odd? Suppose that n is odd. The eigen values/eigenvectors of the skew-circulant matrix A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n are successfully computed in this post. Q. What are ... 2 votes 1 answer 260 views ### The eigenvalues of the matrix \Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n What are the eigenvalues/eigenvectors of the matrix A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n when n is odd? 0 votes 0 answers 39 views ### What information concerning the eigen-structure are transformed on the antidiagonal submatrices? Let us fix symmetric matrices A_1 A_2 in M_m(\mathbb{R}) with A^2_1=\alpha I and A^2_2=\beta I for some positive \alpha ,\beta. For a given matrix B\in M_m(\mathbb{R}), let us ... 1 vote 2 answers 129 views ### Transforming matrix to off-diagonal form I wonder if one can write the following matrix in the form A = \begin{pmatrix} 0 & B \\ B^* & 0 \end{pmatrix}. The matrix I have is of the form$$ C = \begin{pmatrix} 0 & a & b & ... 7 votes 0 answers 117 views ### Measurability of eigenvalues-eigenvectors of a positive compact operator Let$H$be a separable Hilbert space over$\mathbb{R}$. Let${A} = \{a\colon H\to H\,|\,a\text{ is a positive, compact linear operator}\}$. By the spectral theorem, given$a \in A$, there are scalars$...
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}=...