# Questions tagged [eigenvalues]

eigenvalues of matrices or operators

779
questions

7
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0
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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 ...

3
votes

0
answers

64
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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. ...

-5
votes

0
answers

47
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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 ...

0
votes

0
answers

49
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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 ...

2
votes

1
answer

100
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### Is the sum of the circulant matrix with a super upper triangular matrix diagonalizable?

By the circulant matrix $C$ in $M_n(\mathbb{R})$, we mean that
$$C=[e_n|e_1|\cdots|e_{n-1}]$$ where $e_1,\cdots,e_n$ are the standard basis vectors in $\mathbb{R}^n$. It is well-known that
$$C=\...

1
vote

0
answers

82
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### Eigenvalues/eigenfunctions of a diffusion generator

Consider the following symmetric second order diffusion operator, defined, for $\phi \in \mathcal{C}^{2,1}_c\left(\mathbb{R}\times \mathbb{R}_+\right)$, by:
$$L\phi := \lambda_1 \partial_{R_1}(R_1 \...

2
votes

0
answers

51
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### Maximizing the first Neumann eigenvalue on disks

Let $D^2$ be a smooth disk and for any Riemannian metric in $D$, let $\mu_1(g)$ be the first positive Neumann eigenvalue of the Laplacian on $(D, g)$. Li and Yau proved that
$$\mu_1(g) \operatorname{...

0
votes

0
answers

43
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### Computing the eigenvalues of $A+E$ where $A$ is an upper triangular matrix whose diagonal entries are all zero and $E$ is a rank one matrix

Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ ...

7
votes

0
answers

107
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### Steklov eigenvalue for circle valued functions

Let $(M,g)$ be a compact Riemannian manifold with boundary. It is well known that the first positive Steklov eigenvalue $\sigma_1$ of $M$ has the following variational characterization:
$$\sigma_1(M,g)...

0
votes

0
answers

54
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### Construct a permutation matrix from some eigenvectors and eigenvalues

Given $n$ orthonormal vectors $v_1, \dots, v_n \in \mathbb R^d$, where $d > n$, we can show that there are many orthogonal matrices $X$ of size $d$ such that $v_1, \dots, v_n$ are eigenvectors of $...

1
vote

1
answer

58
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### Directed graph whose adjacency matrix admits only 0 as eigenvalue

Let $G$ be a directed graph and let $P_i$
be its vertices. Let $A$
be the corresponding adjacency matrix of $G$, i.e. $a_{i,j}=1$
if and only if there is a directed edge from $P_i$
to $P_j$, ($a_{i,...

10
votes

2
answers

1k
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### Is the eigenvalue map open?

The eigenvalue map in question is
$\sigma: {\mathfrak gl}(\mathbb{C}, n) \to S_n \backslash \mathbb{C}^n$,
from $n$ by $n$ complex matrices to $\mathbb{C}^n$ vectors modulo permutation of entries by $...

0
votes

1
answer

101
views

### Eigenvalues of $\operatorname{diag}({\bf v}) - {\bf v} {\bf v}^\top - \alpha({\bf v} - {\bf w})({\bf v} - {\bf w})^\top$

Given vectors ${\bf v}, {\bf w} \in [0,1]^n$ , where $n \in \mathbb{N} \setminus \{0\}$, and $\alpha > 0$, I would like to find the eigenvalues of the following matrix.
$$\operatorname{diag}({\bf v}...

1
vote

0
answers

48
views

### Fastest algorithm for finding the closest semi-definite matrix?

Given a real-valued, symmetric matrix $A \in \mathbb{R}^{n \times n}$, I'm interested in finding the closest positive semi-definite matrix $X^*\in \mathbb{R}^{n \times n}$:
$$
X^* = \mathop{\text{...

5
votes

2
answers

636
views

### Matrices with same eigenvalues

This question is a more precise version of this question.
Let's assume we have the matrix
$$\left(
\begin{array}{ccccc}
0 & a & 0 & 0 & 0 \\
f & 0 & b & 0 & 0 \\
0 &...

3
votes

1
answer

118
views

### Eigenvalues two-fold degenerate

Consider the matrix $$A:=\left(
\begin{array}{cccc}
0 & a & 0 & 0 \\
f & 0 & b & 0 \\
0 & e & 0 & c \\
0 & 0 & d & 0 \\
\end{array}
\right)$$
I ...

2
votes

0
answers

61
views

### Spectrum of 'complexified' Laplace operator

Let $(M^n,g)$ be a closed Riemannian manifold. Let $\Delta$ be the Laplace–Beltrami operator acting on scalar functions defined on $M$, and let
$\lambda_1 < \lambda_2 \leq \cdots$ be its spectrum.
...

1
vote

0
answers

40
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### QR algorithm for eigenvalues and eigenvectors of large symmetric matrices

I am trying to write a QR algorithm in Python for eigenvectors and eigenvalues finding for large symmetric matrices,
My initial thought was to use Householder transformation with a Wilkinson shift ...

1
vote

1
answer

112
views

### Matrix transformation that always works?

Consider the matrix
$$A_2:= \begin{pmatrix} a & b_1 \\ b_2 & a\end{pmatrix}.$$
Let $\sigma_2 = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}$, then
$$\sigma_2 A_2 \sigma_2 = \begin{...

3
votes

1
answer

250
views

### 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 ...

1
vote

0
answers

55
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### Eigenvalue decomposition of normalized adjacency matrix

Let $A$ be an adjacency matrix of undirected graph $G$, where $G$ is a connected graph. The normalized adjacency matrix is defined as $\hat{A}=D^{-1/2}AD^{-1/2}$, where $D$ is degree matrix of graph $...

0
votes

0
answers

50
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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) = ...

1
vote

0
answers

47
views

### 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 ...

3
votes

1
answer

91
views

### Maximum norm within a random subspace intersected with an ellipsoid

Let $d < n$, and let $G_n(d)$ denote the space of all $d$-dimensional subspaces of $\mathbb{R}^n$.
Let $a = (a_1,\dots, a_n)$ denote a positive sequence, and define
$U(a) = \{u \in \mathbb{R}^n: \...

0
votes

0
answers

29
views

### $\min(|\lambda_{\min}(A(c))|)$ for a special matrix $A(c)$ defined over $\{-1,-1\}^N$

For a given constant $E$, is there way to find the lower bound of the following expression?
$\min_{c\in\{-1,+1\}^N, -\sum_{i,j}c_ic_j=E}(|\lambda_{\min}(A(c))|)$ for matrix $A(c)$ defined over $\{-1,-...

0
votes

1
answer

101
views

### 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 ...

1
vote

0
answers

71
views

### Higher dimensional Cauchy interlacing theorem

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
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### 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 $...

0
votes

0
answers

140
views

### Finding the eigenvectors of a submatrix

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}=...