20
votes
Differentiability of Eigenvalues - Perturbation Theory
The complete reference is Kato's book Perturbation theory .... But perhaps you need only the most basic results. Then see my book Matrices (Springer GTM #216), 2nd edition. This is Section 5.2.
Mind ...
- 52.3k
15
votes
Accepted
Eigenvalues of the complement of a graph
Edit (bis). There are two answers, depending on whether loops about vertices are allowed or not. In addition, the case of regular graphs is completely described.
If loops are allowed
The relation ...
- 52.3k
13
votes
Accepted
What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix?
The $2^n\times 2^n$ dimensional Hadamard matrices $H_{2^n}$ are also called Sylvester matrices or Walsh matrices. There are only two distinct eigenvalues $\pm 2^{n/2}$, so the eigenvectors are not in ...
- 188k
10
votes
Lower eigenvectors of nonnegative matrices with zero trace
(1) No. Counterexample: the symmetric $3 \times 3$ matrix
$$
M(a,b) = \left[
\begin{array}{ccc} 0 & a & b \cr a & 0 & b \cr b & b & 0 \end{array}
\right]
$$
with $0 < b &...
- 79.9k
10
votes
How to show the following matrix has eigenvalues $-d,-d+1,...,d$?
You hope to show for each $n$ that the $(n+1)\times(n+1)$ matrix
$$
B_n = \begin{pmatrix}
0 &n & 0 &0 & \cdots &0 & 0 \\
1 & 0 & n-1 &0& \cdots & 0 & ...
- 3,054
9
votes
Differentiability of Eigenvalues - Perturbation Theory
Eigenvalues are roots of the characteristic polynomial. In the case the polynomial has simple roots, they smoothly depend on the coefficients. Below is the proof in the real case. The argument below ...
- 28k
9
votes
Accepted
Eigenvectors of Kronecker Product
Counterexample: the matrix $I \times I$ has eigenvectors that are not in product form, since every vector is an eigenvector of it and not every vector can be written in product form.
- 20.2k
9
votes
Accepted
The eigenvalues of the matrix $\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$
Here, we verify the observation of @BrendanMcKay that the eigenvalues are $n$ with multiplicity $(n+1)/2$ and $-n$ with multiplicity $(n-1)/2$.
Note that your matrix is skew-circulant, and it is known ...
- 656
9
votes
How to show the following matrix has eigenvalues $-d,-d+1,...,d$?
As requested by Iosif Pinelis, here is another version of the (perfectly correct, in my opinion) answer by Kenta Suzuki.
Let $V$ be the $d+1$-dimensional vector space of homogeneous polynomials of ...
- 18.4k
8
votes
Differentiability of Eigenvalues - Perturbation Theory
The eigenvalues of a square matrix $A$ are the roots of the characteristic polynomial, and are analytic except where their multiplicities change.
Thus if (in a certain open region of parameter space) ...
- 54.2k
8
votes
Accepted
Eigenvalues of adjacency matrix of a k-regular graph
If $G$ is regular, then $J$ and $A_G$ are simultaneously diagonalizable (i.e. they have a common set of eigenvectors).
That is, the eigenvalues of $xA_G$ and $J$ (to the same eigenvectors) just add ...
- 13.6k
7
votes
Integer eigenvectors
For an integer matrix $A$ and an integer eigenvalue $\lambda$, $A - \lambda I$ is an integer matrix, and Gaussian elimination will produce rational eigenvectors; multiply by a common denominator and ...
- 54.2k
7
votes
Eigenvalues of a matrix with binomial entries
I don't have an answer, but it appears that the eigenvalues are always real. I don't have a proof, but have checked this using Sturm sequences for $1 \le a \le b \le 30$.
You're unlikely to get "an ...
- 54.2k
7
votes
Constructive proof of a rational version of Perron-Frobenius?
A course in constructive algebra by Mines, Richman & Ruitenburg, covers the constructive development of discrete fields and their algebraic completion, and even their valuation-metric completion.
...
- 1,616
7
votes
What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix?
It seems to me that $H_{N}$ is the character table of an elementary Abelian $2$-group of order $2^{n}$ (with respect to a suitable ordering of elements). As such, its rows are orthogonal by the ...
- 44.4k
7
votes
Accepted
Continuity of eigenvectors
Yes. Let the size of your matrix be $n$.
Your condition implies that there is an $n-1\times n-1$ submatrix whose determinant is not identically equal to $0$.
Assume without loss of generality that ...
- 91.8k
6
votes
Accepted
Lower bound on the entries of the Perron vector
This seems to be answered in the accepted answer to this question: The height of the Perron-Frobenius eigenvector
For convenience, here is the estimate:
- 96.4k
6
votes
Accepted
Can a vector-function $v:\mathbb{R}^n\to \mathbb{R}^n$ be an eigenvector of its own Jacobian matrix?
I am surprised that Ramesh's answer was voted up. Its first paragraph does not convince me. Here is my analysis:
Consider an integral curve of $v$, that is a curve $t\mapsto X(t)$ so that $\dot X=v(X)...
- 52.3k
6
votes
Accepted
Eigenvalues of a rank-one update of a symmetric matrix
Given eigenvectors and eigenvalues of the symmetric matrix $A$, you can transform to a basis where $A$ is a diagonal matrix $D$; the vector $x$ in that basis transforms to $\tilde{x}$. Then the ...
- 188k
6
votes
Accepted
Given an eigenvalue equation (elliptic PDE) in a ball $B_R$, prove the convergence of the first nonzero $\lambda_R$ and its eigenfunction $\phi_R$
For simplicity let me take $H$ smooth or at least $C^\alpha$ so that I don't have to worry about elliptic estimates:
Limit as $R\to\infty$:
Use
$$
\lambda_R = \inf_{\phi \in C^\infty_c(B_R)}\frac{\...
- 7,197
5
votes
Accepted
Constructive proof of a rational version of Perron-Frobenius?
Yes, both Theorem 1 and Theorem 2 have constructive proofs.
In the following, I will work with rational numbers, but the same arguments
work for any ordered field. (Note that in constructive logic, ...
- 33.8k
5
votes
Accepted
Eigenvectors of a matrix with entries involving combinatorics
In fact for a fixed $n$, the matrices $M(l, n)$ for $l>0$ commute with each other and thus are simultaneously diagonalisable. For your second question, if $\{p_j(y)\}$ is a sequence of polynomials ...
- 383
5
votes
The existence of $v\in A\otimes_{\mathbb{K}}A$ such that $(a\otimes_{\mathbb{K}}1)v=(1\otimes_{\mathbb{K}}a)v$
Let $I$ be the ideal in $A\otimes A$ generated by elements of the form $a\otimes 1-1\otimes a$. (Equivalently, this is the kernel of the multiplication map $A\otimes A\to A$.) We will assume that $0\...
- 56.9k
5
votes
Accepted
Find a way to apply the MLE on Fisher or Covariance matrix to make cross-correlations
OK, let's turn the tables around. I'll describe in my language a physical setup, the mathematical formulation, and the answer that fit the three words I understand: "detector", "error&...
- 61.9k
5
votes
Accepted
Directed graph whose adjacency matrix admits only 0 as eigenvalue
$0$ is the only eigenvalue of $A$ if and only if $A$ is nilpotent, which is equivalent to $A^n=0$. But $A^n=0$ expresses that for all $i$ and $j$, there is no path of length $n$ from $P_i$ to $P_j$. ...
- 22.5k
4
votes
Accepted
Eigenvector of a nonnegative matrix in closed form
If you write $A\nu=\nu$ as a system of equations it can be written as
$$
\frac{\nu_i}{1-\alpha_i}=\sum_{j=1}^{n}\frac{\alpha_i\nu_j}{1-\nu_j}
\qquad \forall i=1, ..., n.
$$
Then the change of ...
- 131
4
votes
Accepted
Solving linear system when one eigenvalue is known
Yes and no.
If you modify slightly the implementation of a Krylov subspace method such as GMRES, you can construct a method with a projection subspace that contains the known eigenvector. This ...
- 20.2k
4
votes
"Unimodality" of the positive eigenvector of a non-negative irreducible matrix?
Let $A$ be as in the question and let $n$ denote the size of $A$, i.e. $A \in \mathbb{R}^{n \times n}$. Troughout, fix $i^* \in \{1,\dots,n\}$. We call a vector $x \in \mathbb{R}^n$...
... $i^*$-...
- 12.5k
4
votes
Differentiability of eigenvalue and eigenvector on the non-simple case
I think Theorem 6.8 on page 122 in Kato: Perturbation Theory for Linear Operators may help (at least for the question concerning the eigenvalues of the symmetric $A$ and $B$ matrices).
Theorem:
...
- 431
4
votes
XOR circulant matrices?
The object in question is also known as the group matrix or the Dedekind matrix, and it is closely related to the Frobenius determinant theorem which was at the origin of the representation theory.
- 1,171
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