All Questions
Tagged with matrices eigenvalues
103 questions with no upvoted or accepted answers
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 ...
- 673
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 ...
- 13.4k
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 $...
- 909
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 ...
- 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
$$\begin{pmatrix}
0 & n-1 & 0 & \dots & 0 \\\
1 & 0 & n-2 & \dots & 0\\\
0 & ...
- 139
8
votes
0
answers
392
views
Bounding eigenvalues by taking high powers of matrices: history?
Let $A$ be real symmetric matrix. It is a well-known observation that we can bound any eigenvalue $\lambda$ of $A$ by using the fact that
$$\lambda^{2 k} \leq \textrm{Tr} A^{2 k}$$
for any $k\geq 1$. ...
- 20.2k
8
votes
0
answers
2k
views
Possible values of eigenvalues of Hadamard product of Hermitian matrices
One of the most important (and very well-known) result in the study of the spectrum of Hermitian matrices is Horn's conjecture (or theorem?), which provides a complete answer to the following problem:
...
- 891
7
votes
0
answers
195
views
Hölder continuity of spectrum of matrices
Endow $\mathbb{C}^{d \times d}$ with the norm induced by the Euclidean norm on $\mathbb{C}^d$. It is well-known (to those who know it well, I guess) that the spectrum $\sigma(A)$ of a matrix $A \in \...
- 12.5k
7
votes
0
answers
905
views
The Möbius function as eigenvalues
Let the $N$ by $N$ matrix $A$ be defined by the tetration:
$$\Large \text{If } \gcd(n, k)=1 \text{ then } A(n,k)= \underbrace{e^{e^{\cdot^{\cdot^{e^{\Re\left(\frac{1}{n^s}\right)}}}}}}_m \text{ else }...
- 1,183
7
votes
0
answers
264
views
Bound on gap between least eigenvalues of $n \times n$ correlation matrix and of its $(n -1) \times (n-1)$ submatrices
The following problem is motivated by one of my research problems.
Let
$\Sigma$ be an $n \times n$ correlation matrix whose least eigenvalue is denoted by $\lambda$.
$\Sigma_i'$ be an ...
- 141
7
votes
0
answers
197
views
A special eigenvalue problem
For my research I need to solve a generalised eigenvalue problem
$Ax=\lambda B x$, where $A$, $B$ are general matrices, and selectively find only eigen-pairs $\lambda, x$ such that $\lambda\in \mathbb{...
- 492
6
votes
0
answers
396
views
Typical eigenspectrum of a random projection of a large matrix
Suppose I have a real symmetric $m \times m$ matrix $\Lambda$. This matrix is large ($m \gg 1$) and, for simplicity, we'll assume it's diagonal. I then construct a random $n \times n$ projection
$$ A =...
- 453
6
votes
0
answers
96
views
Finding the maximal component of a vector in sublinear time
Given a vector $u \in \Bbb R^n$, finding the value of the largest component of $u$ needs linear time in $n$. However, what if we additionally know that $u$ lies in some linear subspace $U \subset \Bbb ...
- 13.6k
6
votes
0
answers
138
views
A question on deformation theory of triples of matrices
Let $(x,y,z)$ be a triple of $n \times n$ traceless complex matrices which are simultaneously diagonalizable. We call such a triple regular if $C_x \cap C_y \cap C_z$ is a Cartan subalgebra of $\...
- 5,215
6
votes
0
answers
465
views
Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits
Let $Sym^2(V)$ be the set of symmetric matrices of a real $n$-dimensional vector space $V$. Given an element $\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where $\lambda_1\leq\...
- 1,452
5
votes
0
answers
327
views
Eigenvalues of Random Regular Bipartite Graphs
I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
- 161
5
votes
0
answers
255
views
Existence of a matrix product from its eigenvalues
Let A and B be two positive definite, real, symmetric matrices. The eigenvalues of A, B and AB, denoted by $\lambda(X)$, obey the relation (from Bhatia):
$$
\lambda^\downarrow(A) \cdot \lambda^\...
- 151
4
votes
0
answers
989
views
Lower bound minimum eigenvalue of a positive semi-definite Hermitian matrix with bounded entries
Let $M \in \mathbb{C}^{n \times n}$ be a matrix with the following properties:
$M$ is Hermitian and positive semi-definite (all the eigenvalues are real and nonnegative).
The diagonal entries of $M$ ...
- 41
4
votes
0
answers
447
views
How to find eigenvalues of following block matrices?
Is there a procedure to find the eigenvalues of A?
$$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t ...
- 41
4
votes
0
answers
284
views
Maximizing a certain eigenvalue ratio
Let $A\in\mathbb{R}^{n\times n}$ be an Hurwitz stable matrix (i.e., the spectrum of $A$ lies on the left-half complex plane) and let $P$ be the unique positive definite solution of the following ...
- 2,712
4
votes
0
answers
2k
views
What is the time complexity of the largest singular value and its vectors?
Full zero-error SVD on an $m \times n$ matrix $A$ would cost $O(\min(m^2n,mn^2))$. What is the time complexity if we need only the largest singular value and its corresponding vectors? I think it is $...
- 123
4
votes
0
answers
84
views
Matrices with almost constant coefficient have a simple eigenvalue
As a by-product of a general result for bounded operators of a Banach space, I have the following:
A matrix $L=(\ell_{ij})_{ij}$ that has almost constant coefficients in the sense that for some $c$,...
- 14.4k
4
votes
0
answers
342
views
Determinant of the sum of a psd (Kronecker) matrix and a diagonal matrix?
Let $K = K1 \otimes K2$ where $K1$ and $K2$ are positive semidefinite matrices. Let $W$ be a diagonal matrix with positive entries. (Everything is real-valued.)
I want to calculate or bound $\det (...
- 141
3
votes
0
answers
145
views
Eigenvalues of random matrices are measurable functions
I have read that if a random matrix is hermitian then its eigenvalues are continuous, hence also measurable.
If the random matrix is not hermitian, the eigenvalues are not continuous in some cases. ...
- 31
3
votes
0
answers
250
views
Generalized matrix determinant lemma for pseudo-determinant of symmetric matrix
The pseudo-determinant of a square matrix $A$ is the product of its nonzero eigenvalues. Consider the generalized matrix determinant lemma $$\det(A+UWV^\top) = \det A\det W\det(W^{-1} + V^\top A^{-1}U)...
- 141
3
votes
0
answers
538
views
Diagonalizing a block tridiagonal matrix
Is there an efficient way to diagonalize a block tridiagonal $N\times N$ matrix of the following form:
\begin{matrix}
A_0 & B & 0 & 0 & \ldots \\
B & A_1 & B & 0 & \...
- 31
3
votes
0
answers
373
views
Eigenvalues of block matrix
Given scalars $\alpha, \beta \in \mathbb{R}$, a symmetric positive definite matrix $A \in \mathbb{R}^{n\times n}$ and a flat matrix $B \in \mathbb{R}^{m\times n}$, where $m < n$, can I say ...
- 125
3
votes
0
answers
498
views
Eigenvectors of sum of SO(3) matrices
I asked this question before on MSE but go no answers. It seems that the problem is rather difficult so I thought of trying here. Given two matrices $A,B\in SO(n)$, each describing a rotation by ...
- 139
3
votes
0
answers
1k
views
Eigenvalues of block-hermitian matrices with zero diagonal blocks
I have a matrix of the form
$$D = \left( \begin{array}{cc} 0 & C \\ C^{\dagger} & 0 \end{array} \right)$$
where $C$ is not necessarily hermitian. In general, can we say anything about the ...
- 47
3
votes
0
answers
220
views
Eigenvalues and eigenvectors of nonsymmetric complex tridiagonal matrix
I wonder if it is possible to find analytically all eigenvalues and eigenvectors of the following $2n \times 2n$ non-symmetric complex tridiagonal matrix
$$M = i \begin{pmatrix}
0 & a & 0 &...
3
votes
0
answers
182
views
Relating Numerical Range and Perron-frobenius theorem for positive matrices?
Let $A$ be any matrix with all entries positive (which means Perron-Frobenius theorem can be applied). Then its numerical range is defined as the set of complex numbers
$$W(A)=\{x^HAx\lvert ~x^Hx=1\}$$...
- 1,421
3
votes
0
answers
221
views
Eigenvalues vs.matrix sparsity
For an n X n matrix whose entries are constrained to be in some [x,y], is the maximum absolute eigenvalue of the matrix a function of its sparsity?
Is there a closed-form expression that states this ...
- 31
2
votes
0
answers
35
views
Limiting spectral distribution of a random matrix with specific structure
First, consider an $N \times N$ Hermitian random matrix $V$ from the Gaussian Unitary Ensemble (GUE). It is well known that the empirical spectral distribution of the GUE satisfies the semicircle law ...
- 21
2
votes
0
answers
85
views
Smallest eigenvalue of certain PD matrix decreases under sparse perturbation
Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...
- 83
2
votes
0
answers
69
views
Unimodular eigenvalue of a H-self-adjoint matrix (indefinite innerproduct)
Let $A,H \in \mathbb{C}^{n \times n}$ be such that $H$ is Hermitian and invertible and $A = H^{-1} A^* H$. In this case, $A$ is said to be $H$-self-adjoint. This is due to the fact that if $\langle \...
- 175
2
votes
0
answers
537
views
Eigenvalues of the sum of matrices, where matrices are tensor products of Pauli matrices
recently I've been studying the toric code (a squared lattice in the context of quantum computation). I want to calculate the energy of the ground state and of all the excitations, with the respective ...
- 21
2
votes
0
answers
121
views
Eigenvalues of two positive-definite Toeplitz matrices
Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are:
$$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)} \qquad M_2[x,y] ...
- 21
2
votes
0
answers
345
views
Extension of the Gershgorin circle theorem for symmetric matrices and localization of positive eigenvalues
In mathematics, the Gershgorin circle theorem can be used to localize eigenvalues of a matrix (including symmetric). Let $A$ be a real symmetry $n × n$ matrix, with entries $a_{ij}$. For $i∈{1,…,n}$ ...
- 145
2
votes
0
answers
81
views
Perturbed Gram matrix
Let $x_t \in \mathbb{S}^{d-1}$, $\forall t\in \mathbb{N}$ and let $e_1$ be the first canonical basis vector of $\mathbb{R}^d$, ie, $e_1 = (1,0,\cdots,0)$. Let us form a Gram Matrix
$$\sum_{t=1}^T(x_t ...
- 215
2
votes
1
answer
392
views
Eigenvalue perturbation under sparse perturbations
Let $A \in \{0,1\}^{n \times n}$ be an irreducible matrix whose entries are in $\{0,1\}$, and let $\lambda_1(A)$ be the eigenvalue with the largest magnitude. By Perron–Frobenius theorem, we know that ...
- 21
2
votes
0
answers
106
views
Connections between eigenvalues of $B$ and $A+iB$
Consider two symmetric and real matrices $A,B\in\mathbb{R}^n$ and definie $A+iB$. Note that $A+iB$ is not hermitian in this case. There are many results based on Brendixson and Courant-Fischer, saying,...
- 21
2
votes
0
answers
146
views
Upper bound on some eigenvalue problem
Let $A_1,\ldots,A_m \in R^{n\times n}$ be symmetric and positive semidefinite, and suppose that their sum $A$ is positive definite. For some nonzero vector $u\in R^n$ with $u^TA_ju>0$ for all $j$, ...
- 3,873
2
votes
0
answers
52
views
Large-scale projected minimum-eigenvalue computations
I am interested in efficient numerical procedures for solving large-scale instances of the following projected minimum-eigenvalue problem:
$$\mu := \min_{v \in \mbox{ker}(A)} \frac{v^T H v}{\lVert v \...
- 35
2
votes
0
answers
613
views
Smallest eigenvalue for Gram matrix of unit norm matrices
Given $n$ symmetric matrices $A_1, \dots, A_n \in \mathbb{R}^{k\times k}$, such that $\|A_i\| \leq 1$ for all $i$, we consider the matrix $M \in \mathbb{R}^{n\times n}$, where $M_{ij} = \langle A_i, ...
- 121
2
votes
0
answers
330
views
Eigenvalues of special sum of Hermitian matrices
In my research on linear algebra and its applications, I have come across the following problem which has stumped me:
Let $ A $ be a positive definite matrix and let $ D $ be a positive diagonal ...
- 620
2
votes
0
answers
550
views
Eigenvalues of a specific Hankel matrix
I have an $\frac{N}{2} \times \frac{N}{2}$ matrix $G$ with entries given by
\begin{equation}
G_{ij} = \frac{1}{\sin(\frac{\pi}{N}(i+j-\frac{3}{2}))}, \;\;\;\;\;\;\;\; 1 \le i,j \le \frac{N}{2},
\end{...
- 101
2
votes
0
answers
677
views
Bounds on smallest Eigenvalue of the Sum of a Standard Laplacian and a Diagonal Matrix
I'm trying to find upper boundaries on the smallest Eigenvalue $\lambda_1$ of $L + E$, where $L$ is a standard Laplacian of an unweighted digraph, with $\lambda_1(L) = 0$ and $E \in \{0,1\}^{n \times ...
- 21
2
votes
0
answers
279
views
Eigenvalues of this matrix
I have a linear map that is defined by $$T:\text{lin}(1,...,x^m) \rightarrow \text{lin}(1,...,x^m) \text{ with}$$ $$x^k \mapsto 2w(k-m)x^{k+1}+(k^2-k-w^2)x^k-2kwx^{k-1}+(k-k^2)x^{k-2}$$
Let me give a ...
- 329
2
votes
0
answers
210
views
Dominant eigenvalue of sum of tridiagonal and diagonal matrices
Suppose I have a tridiagonal square matrix $A$ of some nice form, for which I know the eigenvalues $\lambda_1<\dots<\lambda_n$. $A$ is also essentially nonnegative (nonnegative everywhere except ...
- 21
2
votes
0
answers
132
views
Characterizing the singular values of a matrix with structure
Suppose we have a function from $\mathbb{R}^2\to\mathbb{C}$,
$$f(x,y) = e^{\imath\pi x g(y)}$$
where $g(y)$ is periodic in $y\in[-T, T),\ T<\infty$ (e.g., a sinusoid) and $0\leq x < \infty$
...
- 21