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91 votes
5 answers
124k 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
52 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
  • 12.7k
36 votes
2 answers
32k 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
5k 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
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
  • 13.4k
21 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
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
6 answers
42k 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} & \ddots & \vdots \\\ 0 & b_{2} & a & \...
FlamingWilderbeest's user avatar
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
  • 28.6k
18 votes
1 answer
847 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
  • 3,463
17 votes
3 answers
2k 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
  • 536
16 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
16 votes
2 answers
4k 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
  • 3,217
15 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
  • 153
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
  • 42.8k
15 votes
1 answer
474 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
14 votes
5 answers
989 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
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
  • 2,712
13 votes
0 answers
809 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
  • 673
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
  • 1,549
12 votes
3 answers
737 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
  • 1,324
12 votes
4 answers
904 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
  • 197
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
825 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
  • 13.4k
11 votes
1 answer
927 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
  • 1,269
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
980 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
614 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
5 answers
2k views

The maximal eigenvalue of a symmetric Toeplitz matrix

Let $0\le x\le 1$ be a real number. Denote by $A_n(x)=(a_{ij})$ the $n$ by $n$ matrix such that $a_{ij}=x^{|i-j|}$ and let $\lambda_n(x)$ be the maximal eigenvalue of $A_n(x)$. Is there any ...
Andrei Jaikin's user avatar
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
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
9 votes
3 answers
383 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
9 votes
1 answer
3k views

Frobenius-Perron eigenvalue and eigenvector of sum of two matrices

Suppose that I have two positive matrices, $A$, and $B$, and I know their Frobenius-Perron eigenvalues ($\lambda_A$, $\lambda_B$) and eigenvectors ($v_A$, $v_B$). I'm interested in what I can say ...
sasquires's user avatar
  • 403
9 votes
1 answer
3k views

Connection between eigenvalues of matrix and its Laplacian.

Hello! There are two definitions of graph spectrum: 1) Eigenvalues of adjacency matrix $A$. 2) Eigenvalues of Laplacian of adjacency matrix ($L$). Different sources offer different properties based ...
TotalNoob's user avatar
9 votes
1 answer
396 views

Bound on the ratio of top 2 eigenvalues

Let $P$ be a $n \times n$ stochastic matrix such that $P_{ij}=\tau$ if $i \neq j$ and $P_{ii} = 1 - (n-1)\tau$ where $0<\tau < \frac{1}{n}$. It is clear that the largest eigenvalue of $P$ is 1, ...
user avatar
9 votes
0 answers
802 views

Positive definiteness of matrix

This question is about the positive definiteness of a (non-random) matrix that is defined using random variables as follows: We fix the vector $v=(1,1)$ (yet, it seems the final result does not ...
Kung Yao's user avatar
  • 192
9 votes
0 answers
624 views

Eigenvalues of leading principal submatrix of the Clement-Kac-Sylvester tridiagonal matrix

It's well-known that the eigenvalues of the Clement-Kac-Sylvester tridiagonal matrix $$\begin{pmatrix} 0 & n-1 & 0 & \dots & 0 \\\ 1 & 0 & n-2 & \dots & 0\\\ 0 & ...
Sihuang Hu's user avatar
8 votes
2 answers
15k views

Upper bounds on eigenvalues of PSD matrix?

Suppose A is a symmetric positive semidefinite matrix. Is there a way to upper bound the largest eigenvalue using properties of its row sums or column sums? For instance, the Perron–Frobenius ...
Yisong Yue's user avatar
8 votes
3 answers
1k views

Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?

Suppose that $M$ is an $n \times n$ matrix where each entry is a positive integer. Then $M$ is Perron-Frobenius and so has unique largest real eigenvalue $\lambda_{\textrm{PF}}$. Does an upper ...
Mark Bell's user avatar
  • 3,165
8 votes
3 answers
8k views

Spectrum of an adjacency matrix

The adjacency matrix of a non-oriented connected graph is symmetric, hence its spectrum is real. If the graph is bipartite, then the spectrum of its adjacency matrix is symmetric about 0. A few ...
Delio Mugnolo's user avatar
8 votes
2 answers
12k views

Relation between eigenvalues of $A$ and $A^TA$?

For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$? I ask this because I am looking into the relation between $A$ and $A+cI$, ...
Eric S.'s user avatar
  • 181
8 votes
1 answer
5k views

Eigenvectors of Kronecker Product [closed]

Conjecture If $A$ and $B$ are two complex square matrices, then every eigenvector of $A\otimes B$ is of the form $x\otimes y$, where $x$ is an eigenvector of $A$ and $y$ is an eigenvector of $B$. ...
Henrique de Oliveira's user avatar
8 votes
1 answer
7k views

Upper bound on largest eigenvalue of a real symmetric $n \times n$ matrix with all main diagonal entries positive, everywhere else nonpositive

Is there a good analytic upper bound on the largest eigenvalue of a real symmetric n*n matrix with all main diagonal entries strictly positive, all other entries <=0 with typically many of them ...
equest's user avatar
  • 83
8 votes
2 answers
380 views

Projecting onto space of matrices with spectral radius less than one

Consider the space $$ S = \left\{ A \in \mathbb{R}^{n \times n} : \mathrm{SpectralRadius}(|A|) \leq 1 \right\}$$ where $|A|$ is the entry-wise absolute value. Given a matrix $M \in \mathbb{R}^{n\times ...
CComp's user avatar
  • 123
8 votes
1 answer
904 views

A generalized log inequality for positive definite trace-one matrices

Let $\{V_i\}_{i=1}^N$ be a set of $n\times m$, $n\geq m$, real matrices of full column rank and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. Moreover, let $A^{1/2}=(...
Ludwig's user avatar
  • 2,712
8 votes
0 answers
392 views

Bounding eigenvalues by taking high powers of matrices: history?

Let $A$ be real symmetric matrix. It is a well-known observation that we can bound any eigenvalue $\lambda$ of $A$ by using the fact that $$\lambda^{2 k} \leq \textrm{Tr} A^{2 k}$$ for any $k\geq 1$. ...
H A Helfgott's user avatar
  • 20.2k

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