Questions tagged [eigenvalues]

eigenvalues of matrices or operators

Filter by
Sorted by
Tagged with
86 votes
5 answers
116k views

Eigenvalues of matrix sums

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite? I am ...
Jean-Pierre Gunman's user avatar
51 votes
8 answers
5k views

Is there a fast way to check if a matrix has any small eigenvalues?

I have hundreds of millions of symmetric 0/1-matrices of moderate size (say 20x20 to 30x30) which (obviously) have real eigenvalues. I wish to extract from this list the tiny number of matrices that ...
Gordon Royle's user avatar
  • 12.3k
35 votes
2 answers
30k views

Eigenvalues of the product of two symmetric matrices

This is mostly a reference request, as this must be well-known! Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or $BA$...
kjetil b halvorsen's user avatar
26 votes
1 answer
4k views

Generalization of Cauchy's eigenvalue interlacing theorem?

Cauchy's Interlacing Theorem says that given an $n \times n$ symmetric matrix $A$, let $B$ be an $(n-1) \times (n-1)$ principal submatrix of it, then the eigenvalues of $A$ and those of $B$ interlace. ...
Hao's user avatar
  • 571
23 votes
1 answer
1k views

Eigenvalues of Laplace operator

Assume that $(M,g)$ is a Riemannian manifold. Is there any relation between the sequence of eigenvalues of Laplace operator acting on the space of smooth functions and the sequence of eigenvalues of ...
Ali Taghavi's user avatar
22 votes
4 answers
5k views

Eigenvalues of permutations of a real matrix: can they all be real?

For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and define the total spectrum $TS(M)$ as the union of all their spectra (counting ...
Wolfgang's user avatar
  • 13.2k
21 votes
2 answers
3k views

Integer matrices with no integer eigenvalues

Let $$A = \begin{pmatrix} 3&1 \\ 0&1 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1&0\\ 1&2 \end{pmatrix}$$ I want to show that the only elements of the semigroup generated by $A$ and $B$...
Hej's user avatar
  • 1,045
20 votes
5 answers
2k views

The middle eigenvalues of an undirected graph

Let $ \lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_{2n} $ be the collection of eigenvalues of an adjacency matrix of an undirected graph $G$ on $2n$ vertices. I am looking for any work or references ...
Tomaž Pisanski's user avatar
20 votes
2 answers
2k views

Formula expressing symmetric polynomials of eigenvalues as sum of determinants

The trace of a matrix is the sum of the eigenvalues and the determinant is the product of the eigenvalues. The fundamental theorem of symmetric polynomials says that we can write any symmetric ...
Jules's user avatar
  • 463
19 votes
1 answer
2k views

Smallest eigenvalue of a tricky random matrix

While experimenting with positive-definite functions, I was led to the following: Let $n$ be a positive integer, and let $x_1,\ldots,x_n$ be sampled from a zero-mean, unit variance gaussian. Consider ...
Suvrit's user avatar
  • 28.4k
19 votes
2 answers
559 views

Eigenvalues and eigenvectors of the matrix with entries $\dbinom{n+1}{2j-i}$ for $i, j = 1, 2, \ldots, n$

Let $n$ be a nonnegative integer, and let $B$ be the $n \times n$-matrix (over the rational numbers) whose $\left(i, j\right)$-th entry is $\dbinom{n+1}{2j-i}$ for all $i, j \in \left\{ 1, 2, \ldots, ...
darij grinberg's user avatar
18 votes
5 answers
38k views

Eigenvalues of Symmetric Tridiagonal Matrices

Suppose I have the symmetric tridiagonal matrix: $ \begin{pmatrix} a & b_{1} & 0 & ... & 0 \\\ b_{1} & a & b_{2} & & ... \\\ 0 & b_{2} & a & ... & 0 \...
FlamingWilderbeest's user avatar
18 votes
2 answers
4k views

How are eigenvalues and eigenvectors affected by adding the all-ones matrix?

Given an $n \times n$ matrix $A$ and the $n\times n$ all-ones matrix $J = (1)_{ij}$, I'm interested in the relation between the eigenvalues and eigenvectors of the matrices $A$ and $A+J$, or more ...
Somatic Custard's user avatar
18 votes
1 answer
829 views

Showing that a certain matrix is not positive definite

Let $J_k$ be a $k \times k$ all ones matrix and $B$ any $k \times k$ binary matrix - that is $B$ only has entries from $\{0,1\}$. I would like to show that the matrix $$X_B = (J_k -I) - B (J_k - I)^{-...
Jernej's user avatar
  • 3,433
17 votes
3 answers
1k views

Finding the nearest matrix with real eigenvalues

In this thread on MATLAB Central, I found a discussion on finding the nearest matrix with real eigenvalues. The first hypothesis was to simply truncate the complex part of the eigenvalues. So, given ...
Andrea F.'s user avatar
  • 185
16 votes
2 answers
1k views

Spectral symmetry of a certain structured matrix

I have a matrix $$ A= \begin{pmatrix} 0 & a & d & c\\ \bar a & 0 & b & d \\ \bar d & \bar b & 0 & a \\ \bar c & \bar d & \bar a & 0 \end{pmatrix} $$ As ...
Sascha's user avatar
  • 506
16 votes
2 answers
3k views

The singular values of the Hilbert matrix

The $n\times n$ Hilbert matrix $H$ is defined as follows $$H_{ij} = \frac{1}{i+j-1}, \qquad 1\leq i,j\leq n$$ What is known about the singular values $\sigma_1 \geq \cdots \geq \sigma_n$ of $H$? ...
alext87's user avatar
  • 3,167
15 votes
3 answers
2k views

Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?

The Hermitian matrices form a real vector space where we have a Lebesgue measure. In the set of Hermitian matrices with Lebesgue measure, how does it follow that the set of Hermitian matrices with ...
Guido Li's user avatar
15 votes
1 answer
1k views

Existence of double eigenvalue

Let $A$ and $B$ be complex $4\times 4$ matrices. Assume both are Hermitian, and that they are linearly independent. Must there exist a nonzero real linear combination $aA + bB$ which has a repeated ...
Nik Weaver's user avatar
15 votes
2 answers
6k views

Linearly constrained eigenvalue problem

Suppose I'd like to: \begin{align} \mathop{\text{min}}_\mathbf{x} && \mathbf{x}^T\mathbf{A}\mathbf{x} \\ \text{subject to:} && \mathbf{x}^T \mathbf{M} \mathbf{x} = 1\\ && \...
Alec Jacobson's user avatar
15 votes
2 answers
608 views

Maximum dimension of space of matrices with a real eigenvalue

Let $M_n(\mathbb{R})$ denote the space of all $n\times n$ real matrices. What is the maximum dimension $f(n)$ of a subspace $V$ of $M_n(\mathbb{R})$ such that every matrix in $V$ has at least one real ...
Richard Stanley's user avatar
14 votes
4 answers
7k views

Eigenvectors and eigenvalues of a tridiagonal Toeplitz matrix

Is it possible to analytically evaluate the eigenvectors and eigenvalues of the following $n \times n$ tridiagonal matrix $$ \mathcal{T}^{a}_n(p,q) = \begin{pmatrix} 0 & q & 0 & 0 &...
user22127's user avatar
  • 143
14 votes
5 answers
982 views

Eigenvalues of a matrix with entries involving combinatorics

Let $F(n, l, i, j)$ be the cardinality of the set \begin{eqnarray*} \{(k_1, \cdots, k_n)\in\mathbb{Z}^{\oplus n}|0\leq k_r\leq l-1\text{ for }1\leq r\leq n\text{, }k_1+\cdots+k_n=lj-i\}. \end{eqnarray*...
No_way's user avatar
  • 383
13 votes
3 answers
3k views

Differentiability of Eigenvalues - Perturbation Theory

first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever ...
xmonetx's user avatar
  • 138
13 votes
3 answers
2k views

Eigenvalue pattern

We consider a matrix $$M_{\mu} = \begin{pmatrix} 1 & \mu & 1 & 0 \\ -\mu & 1 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 &-1 & 0 & 0 \end{pmatrix}$$ One easily ...
Dreifuss's user avatar
  • 133
13 votes
2 answers
1k views

A log inequality for positive definite trace-one matrices

Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. I would like to prove (or disprove) the following ...
Ludwig's user avatar
  • 2,682
13 votes
1 answer
431 views

Has Nambu's notion of an "eigenoperator" found a place in the mathematical literature?

The physicist Yoichiro Nambu introduced in a 1950 paper A Note on the Eigenvalue Problem in Crystal Statistics the notion of an "eigenoperator" (page 12, see Nambu and the Ising model for a ...
Carlo Beenakker's user avatar
13 votes
0 answers
702 views

Can one Gershgorin circle (only) contain all eigenvalues, when the other circles are not contained in it

In short, following a question from my students, I am trying to find a special case where all the eigenvalues of a matrix lie within only one circle, but not in the others, and the other circles are ...
Itay's user avatar
  • 661
12 votes
6 answers
2k views

Differentiability of eigenvalues of positive-definite symmetric matrices

Let $A\in M(n,\mathbb{R})$ be an invertible matrix. Consider the (real) eigenvalues $\lambda_1,\cdots,\lambda_n$, in increasing order, of the positive-definite symmetric matrix $A^t A$. We shall ...
Somnath Basu's user avatar
  • 3,403
12 votes
2 answers
1k views

Eigenvalue perturbation theory via Feynman diagrams

Suppose I have a matrix given by a sum $$A=D+\epsilon B$$ where $D$ is diagonal and $\epsilon$ is small, and I want the eigenvalues of $A$ as a power series in $\epsilon$. The first two orders in ...
thedude's user avatar
  • 1,417
12 votes
3 answers
699 views

How to find Suleimanova's work on the Nonnegative Inverse Eigenvalue Problem?

Many papers cite the work of Suleimanova when studying inverse eigenvalue problems - in particular, the nonnegative inverse eigenvalue problem (NIEP). However, I cannot seem to find her work anywhere....
Jeremy Lin's user avatar
12 votes
2 answers
2k views

What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix?

What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix defined recursively by $H_1=(1)$ and $$ H_N=\begin{pmatrix}H_{N/2} & H_{N/2} \\ H_{N/2} & -H_{N/2}\end{pmatrix}, $$ ...
MCH's user avatar
  • 1,304
12 votes
4 answers
823 views

Show that the eigenvalues of a non-symmetric matrix built from positive matrices have positive real parts

Let $A, B, C \in \mathbb{R}^{n\times n}$ such that $N = \begin{bmatrix} A & B\\ B^{\top} & C\end{bmatrix}$ is a symmetric positive definite matrix. I'm trying to show that the following matrix ...
PAb's user avatar
  • 187
12 votes
1 answer
1k views

Eigenvalues come in pairs

Consider the two matrices with some parameter $s \in \mathbb R$ $$A_1= \begin{pmatrix} s& -1 &0& 0 \\1&0 &0&0 \\ 0&0&1&0 \\0&0&0&1 \end{pmatrix}$$ and $$...
Pritam Bemis's user avatar
12 votes
0 answers
208 views

Classes for which the Spectrum determines a Convex Shape

Given a planar domain $\Omega \subset \Bbb{R}^2$ bounded and open we can associate to it the spectrum of the Laplace operator with Dirichlet boundary condition. It is known that there are planar ...
Beni Bogosel's user avatar
  • 2,102
12 votes
0 answers
816 views

Eigenvalues of permutations of a real matrix: how complex can they be?

This is sort of complementary to this thread. I’ll repeat the definitions here: For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and ...
Wolfgang's user avatar
  • 13.2k
11 votes
2 answers
1k views

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 $...
Lucas Seco's user avatar
  • 1,103
11 votes
8 answers
2k views

Semicircle law universality elsewhere

Wigner's semicircle distribution is: $$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$ Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows ...
Alex R.'s user avatar
  • 4,902
11 votes
1 answer
908 views

Imaginary eigenvalues

Consider the matrix $$A(\mu) = \begin{pmatrix} 0 & 1& 0 & 0 \\ -1 & -i\mu & 0 & i \\ 0 & 0 & 0 & 1 \\ 0 &i & -1 & i\mu \end{pmatrix}.$$ This matrix is ...
Pritam Bemis's user avatar
11 votes
1 answer
2k views

Eigenvalues of the complement of a graph

Let $A$ and $\widetilde A$ be the adjacency matrices of a graph $G$ and of its complement, respectively. Is there any relation between the eigenvalues of $A + \widetilde A$ and the eigenvalues of $A$ ...
GA316's user avatar
  • 1,219
11 votes
3 answers
1k views

Maximum singular value of a random $\pm 1$ matrix

Define a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ such that each element is independently and randomly chosen with probability $\frac 12$ to be either $+1$, or $-1$. Do you know any result in ...
Kostas's user avatar
  • 199
11 votes
1 answer
865 views

Exact eigenvalues of a specific tridiagonal matrix

I'm studying the following tri-diagonal matrix $$ X = \begin{pmatrix} 0 & x_0 & 0 & 0 &\cdots & 0 & 0 & 0 \\\ x_0 & 0 & x_1 & 0 &\cdots & 0 & ...
Kasper's user avatar
  • 161
10 votes
2 answers
580 views

Lower eigenvectors of nonnegative matrices with zero trace

Let $A$ be an $N\times N$ nonnegative matrix with all diagonal entries equal to zero and such that there is $n_0$ such that all entries of $A^{n_0}$ are strictly positive. Let $\lambda_1,\ldots, \...
Serguei Popov's user avatar
10 votes
2 answers
448 views

Does approximate equality of quantum states imply operator inequality in a large subspace?

Let the trace norm of $X$ be $$\Vert X\Vert_1 := \operatorname{tr} \left(\,(X^\dagger X)^{1/2}\right)$$ and let the operator inequality $A \leq B$ denote that the operator $B-A$ is positive ...
Noel's user avatar
  • 165
10 votes
1 answer
1k 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 \...
Jefflee's user avatar
  • 103
10 votes
2 answers
452 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 &...
Kung Yao's user avatar
  • 192
10 votes
0 answers
237 views

Generalized eigen property of a matrix

Given a $n \times n$ invertible matrix $A$, I am interested in the set $$ \mathcal{S}(A) = \{ D \textrm{ diagonal matrix } \mid \det(D - A) = 0 \}. $$ Thus, for all eigenvalues $\lambda_i$, we have $...
Jiro's user avatar
  • 909
10 votes
0 answers
228 views

Maximum dimension of a space of $n\times n$ real matrices with at least $k$ nonzero eigenvalues

Let $M_n(\mathbb{R})$ denote the $n^2$-dimensional real vector space of real $n\times n$ matrices. Let $\rho_k(n)$ denote the maximum dimension of a subspace $V$ of $M_n(\mathbb{R})$ such that every ...
Richard Stanley's user avatar
9 votes
3 answers
2k views

What happens to eigenvalues when edges are removed?

I am stuck at the following : Let $G$ be a graph and $A$ is its adjacency matrix. Let the eigenvalues of $A$ be $\lambda_1\le \lambda_2\leq \cdots \leq \lambda_n$. If we remove some edges from the ...
Charlotte's user avatar
  • 444
9 votes
3 answers
378 views

convergence of 2nd eigenvalue

Fix $0<h_1<h_2<h_3<1$ reals. All matrices below are $3\times3$ real. Suppose the sequence of matrices $M(n)$ are symmetric positive definite and these converge (point-wise) to a symmetric ...
T. Amdeberhan's user avatar

1
2 3 4 5
17