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Results tagged with matrices
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user 1266
Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.
8
votes
Connection between eigenvalues of matrix and its Laplacian.
Essentially your question is equivalent to asking for the relation between the spectrum of $A+D$ and $A$, where are $A$ is symmetric, $D$ is diagonal and both matrices are real. …
- 12.1k
9
votes
Accepted
Spectrum of an adjacency matrix
Since the eigenvalues are real, and since their sum is the trace of $A$, which is zero, we see that either all eigenvalues are zero, or there are both positive and negative eigenvalues. So no non-empt …
6
votes
Integral roots of circulant matrix
First, this is due to Bridges and Mena ``Rational G-matrices with rational eigenvalues'', J. Comb. Theory A, (1982), 264-280,. …
- 12.1k
3
votes
Are there non-isomorphic graphs with rationally orthogonal similar adjacency matrices?
Yes, there are. The smallest pair are the subdivision graph of $K_{1,3}$ and $C_6$ with an isloated vertex. There are many more, look up "Godsil-McKay switching".
The original source is at http://cs. …
- 12.1k
10
votes
Accepted
When are the adjacency matrices of non-isomorphic graphs similar?
There is no characterization known of when a graph is determined by its spectrum. The probability that a tree on $n$ is determined by its characteristic polynomial goes to zero as $n$ tends to infinit …
- 12.1k
1
vote
Eigendecomposition of a summation of matrices
There is nothing useful to say. Consider the decomposition $I=A+(I-A)$. There is no useful relation between the eigenvalues of $I$ and those of $A$.
- 12.1k
5
votes
Which directed graphs have a normal adjacency matrix?
(In fact the obvious variant of the proof for
adjacency matrices works.)
On five vertices, my sage calculations found 111 balanced directed graphs from a
total of 9608. …
- 12.1k
21
votes
Eigenvalues of symmetric tridiagonal matrices
Amongst the polynomials that can arise as characteristic polynomials of tridiagonal matrices with zero diagonal, one finds the Hermite polynomials. …
- 12.1k
7
votes
Accepted
Non symmetric matrices with real eigenvalues
$$
\begin{pmatrix}1&0\\\0&k^{-1/2}\end{pmatrix}
\begin{pmatrix}A_1&A_2\\\ kA_2^T&A_3\end{pmatrix}
\begin{pmatrix}1&0\\\0&k^{1/2}\end{pmatrix}
= \begin{pmatrix}A_1& k^{1/2}A_2\\\ k^{1/2}A_2^T&A …
- 12.1k
4
votes
Full-rank linearly independent matrices
If the characteristic of $\mathbb{F}$ is not two and $E_{i,j}$ $(1\le i,j\le n)$ is the "standard basis", then the matrices $I+E_{i,j}$ are invertible. …
- 12.1k
5
votes
Repeated Second Eigenvalue of the Adjacency Matrix of a Graph
It you allow weighted adjacency matrices and if you insist (among there things) that the eigenspace associated to $\lambda_2$ satisfies the "strong Arnold condition", then you are dealing the Colin de …
- 12.1k
7
votes
Accepted
Calculating the Perron-Frobenius eigenvector of a positive matrix from limited information
1.579071, 1.460275, 1.019079, 1.168003, 0.5330099, 0.206667, 0.612263)
$$
for $H_2$ and, for $H_5$,
$$
(1, 1, 1.579071, 2.0725388, 1.631342, 2.134811, 0.974205, 0.377735, 0.827744)
$$
If you want positive matrices …
- 12.1k
3
votes
Computing the multiplicity of an eigenvalue of a 0-1 symmetric matrix...
I suspect there are few short cuts in general and that computing multiplicities for 01-matrices
will not be easier than computing them for real symmetric matrices. …
- 12.1k
6
votes
2-norm of the upper triangular "all-ones" matrix
[This may be largely an alternate version of Noam's answer, but the extra context could be
interesting.]
Let $N$ be the $m\times m$ matrix with $N_{i,i+1}=1$ for $i=1,\ldots,m-1$
and all other entri …
- 12.1k
14
votes
Accepted
Eigenvalues of the sum of a diagonal and a unit matrix
Recall that $\det(I-AB)=\det(I-BA)$ for any matrices
$A$ and $B$ such that both products $AB$ and $BA$ are defined. …
- 12.1k