All Questions
Tagged with matrices sp.spectral-theory
77 questions
0
votes
1
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204
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Are these particular kinds of matrices well known?
Given two positive integers $n$ and $a \leq \frac{n}{2}$ consider a $n \times n$ matrix $A$ such that,
all the diagonal entries are either $a$ or $a+1$
all the non-zero off-diagonal entries are $\pm ...
- 1,893
4
votes
5
answers
4k
views
About adding a negative definite rank-1 matrix to a symmetric matrix
If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$)
I guess that the eigenvalues of $B - vv^T$ ...
- 1,893
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
7
votes
1
answer
1k
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Origins of the Jacobi matrix
I have several questions concerning history of Jacobi matrices.
Does anybody know why the Jacobi matrix (=symmetric tridiagonal matrix) is named by Carl Gustav Jacob Jacobi? What was his contribution ...
- 2,188
1
vote
1
answer
546
views
Existence of a real eigenvalue
I have a matrix $M \in \mathbb{R}^{(n+1) \times (n+1)}$ that is tridiagonal.
In numerical computations I found out that I always find a real eigenvalue. My question is: Is there a theorem that ...
user37929
3
votes
1
answer
944
views
numerical range of a column-zero-sum matrix
I am trying to produce an example of a (necessarily non-normal) matrix that has only eigenvalues with positive real part, but whose numerical range contains elements with strictly negative real part. ...
- 3,388
1
vote
1
answer
720
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Eigenvalues of Sum of non-singular matrix and diagonal matrix
Suppose $D={\rm diag}(d_i)$ is a diagonal matrix with all diagonal entries $d_i=\pm 1$. This implies $D^2=I$.
Suppose $A$ is a non-singular Hermitian matrix. If we know that $A+A^{-1}+D$ has rational ...
- 427
5
votes
2
answers
1k
views
spectral radius monotonicity
I encountered an inequality when reading a paper. Can someone help to show how to prove it?
Let be the spectral radius of matrix $A$ or $\rho(A)=\max\{|\lambda|, \lambda \text{ are eigenvalues of ...
- 2,239
2
votes
0
answers
426
views
eigenvalues of the sum of a stochastic matrix and a diagonal matrix
Let $D$ be a real diagonal matrix $D=diag(a_1,a_2,\ldots,a_n)$ with $a_1\le a_2\le\ldots\le a_n$. Assume that at least one of the $a_i$ is positive. Let $P$ be an irreducible, real, row-stochastic ...
- 21
12
votes
6
answers
692
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Invertibility of a certain matrix indexed by the Hamming cube
For reasons which the margin of this page is too small to hold, I have been reading parts of a recent paper by O. Selim
On submeasures on Boolean algebras, arXiv 1212.6822v3
and in Section 7 the ...
- 25.8k
2
votes
1
answer
2k
views
Eigenvalues of directed Laplacian matrix $L$ and $DL$, where $D$ is a diagonal matrix with positive entries
I have a weighted Laplacian matrix $L$ of a strongly connected directed graph and a diagonal matrix $D$ with positive entries. Since the graph is directed, $L$ is non-symmetric real. Further, since ...
- 21
2
votes
1
answer
276
views
Asymptotic Behavior of Non-Analytic Function of the Eigenvalues
Hello,
Let $A_n = (a_{k-j};\;k,j = 0,1,\ldots,n-1)$ be a sequence of $n\times n$ Toeplitz matrices, with eigenvalues $(\lambda_{n,i};\;i = 0,1,\ldots,n-1)$.
If $A_n$ were a sequence of Hermitian ...
- 43
1
vote
2
answers
18k
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What does multiplying a matrix by its transpose have to do with spectral theorem? [closed]
What does multiplying a matrix by its transpose have to do with spectral theorem? I basically am trying to understand what this would mean with regards to spectra of waves.
I think it give you a ...
- 19
5
votes
3
answers
271
views
Conditions ensuring an order betweenthe smallest eigenvalues of two positive definite Jacobi matrices
Let $J, L$ be two symmetric positive definite tridiagonal matrices of positive diagonal entries, $\mbox{diag}(J)=(a_1, a_2, \ldots, a_n)$, $\mbox{diag}(L)=(\alpha_1, \alpha_2, \ldots, \alpha_n)$, ...
- 143
7
votes
1
answer
271
views
Singular values of $X+iY$ where $X$ and $Y$ are Hermitian
Lets have two Hermitian $n\times n$ matrices $X$ and $Y$.
Are there any known properties of the singular values of
$$Z = X + i Y.$$
I am the most interested in bounding from above a few first ...
- 1,612
8
votes
0
answers
738
views
Bounding sum of first singular values squared for Kronecker sum of traceless matrices
Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e.
$$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + B^\...
- 1,612
9
votes
1
answer
1k
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0 eigenvalue for a symmetric tridiagonal matrix
Let $T\in \mathbb{R}^{n\times n}$ be a symmetric tridiagonal matrix having the off--diagonal entries equal to -1. The diagonal entries are all positive, $a_i>0$, $i=\overline{1,n}$, and there ...
- 143
3
votes
1
answer
284
views
Estimating spectral radius with a Gaussian vector
Suppose I'm trying to estimate the spectral radius of a square $n \times n$ matrix $A$,
and let $N$ be a distribution over Gaussian i.i.d. vectors of length $n$.
Is the following lemma true:
If the ...
- 33
1
vote
0
answers
227
views
Joint Convexity of Spectral functions of several matrices
$\{A_1 \ldots A_K \}$ is a set of matrices in $\mathbb{R}^{m \times n}$. Let $f (A_1,\ldots,A_K)$ be a function of the singular values of all matrices. For e.g., $f$ is just summation of singular ...
- 519
17
votes
5
answers
2k
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Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations?
This question is related to another question, but it is definitely not the same.
Is it always possible to diagonalize (at least locally around each point) a family of symmetric real matrices $A(t)$ ...
- 4,312
4
votes
1
answer
1k
views
dominant eigenvector
Hi, everyone! Is there any efficient way to simplify the following tensor product
$X \otimes X + X^T \otimes X^T$, where $X$ is a square $n \times n$ matrix.
My goal is to efficiently compute the ...
- 41
20
votes
2
answers
8k
views
Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters
Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb R^k\...
- 4,312
29
votes
3
answers
3k
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Perron-Frobenius "inverse eigenvalue problem"
The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...
- 2,200
3
votes
3
answers
3k
views
Infinite hermitian matrix
Suppose we have a finite square n x n matrix of complex numbers H that is Hermitian and skew-symmetric:
$H^\dagger = H$ and $H^T = -H$.
(T denotes transpose, $\dagger$ denote conjugate transpose. I ...
- 195
5
votes
1
answer
600
views
Spectrum of a generic integral matrix.
My collaborators and I are studying certain rigidity properties of hyperbolic toral automorphisms.
These are given by integral matrices A with determinant 1 and without eigenvalues on the unit circle....
- 4,188
5
votes
0
answers
539
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An inverse eigenvalue problem on Jacobi matrices
I am interested in trying to design a Hermitian Jacobi (tridiagonal) matrix $H$ that has specific properties. The basic property, which is simple enough to construct, is that for an $N\times N$ matrix ...
- 291
1
vote
2
answers
876
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Matrix logarithms are not unique
In my ODE class, we proved that if $\exp(L) = \exp(L')$ then the eigenvalues are congruent mod $2 \pi i$. Here, $L$ and $L'$ are two $n \times n$ matrices. I wanted to know if something more precise ...
- 22.8k