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Results tagged with matrices
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user 9025
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.
1
vote
Accepted
Methods to solve for a matrix whose entries satisfy certain properties
There are zero or infinitely many solutions depending on where the non-zero entries have to be. So there is no general-purpose answer.
I don't think your equations are properly stated, as $\boldsymbol …
- 37.7k
1
vote
Which directed graphs have a normal adjacency matrix?
For all 0-1 matrices, rather than isomorphism classes, the numbers are in https://oeis.org/A055547 . …
- 37.7k
15
votes
Accepted
Why does an invertible complex symmetric matrix always have a complex symmetric square root?
Higham, in Functions of Matrices, Theorem 1.12, shows that the Jordan form definition is equivalent to a definition based on Hermite interpolation. …
- 37.7k
2
votes
Non-singular matrix with restricted entries
[EXPANDED]
PART 1 (also done by Peter)
If $x,y$ are coprime and have opposite sign, there is a singular symmetric matrix with 1 on the diagonal and
only $x$ and $y$ off the diagonal.
Say $x<0,y>0$. Si …
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2
votes
Accepted
A combinatorial matrix reconstruction problem
As Carlo proposed in a comment, I'll take it that we have an unordered list of $n$ unordered lists of $n$ elements, and we want to identify the symmetric matrices for which those $n$ lists correspond to … $ is any symmetric matrix).
$$
\pmatrix{0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & M}
\qquad
\pmatrix{1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & M}
$$
The two matrices …
- 37.7k
2
votes
How to generate constant row and column sum matrices?
This is an example of generating a random point in a polytope. There is some work on that general problem, but I don't know if it would be practical for your special case. For example:
Rubin, Generat …
- 37.7k
4
votes
Relation graph isomorphism to discrete logarithm
The expression of the problem in terms of graphs or 0-1 matrices is a red herring and I suggest a more natural formulation is as follows. … In the other direction, consider matrices of order $|\varOmega\times\varOmega|$ with only zeros off the diagonal.
To solve it, use Joseph's approach. …
- 37.7k
2
votes
Accepted
The eigenvectors and eigenvalues of Laplacian matrix in a chain graph
You can find it in this paper.
In case that link doesn't work, search at Google Scholar for "On the observability of path and cycle graphs".
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8
votes
Polynomial time algorithm for rigid graph isomorphism
You have reduced the graph isomorphism problem to a 0-1 programming problem. 0-1 programming problems are NP-hard in general, so the question is whether your particular case is an exception. You haven …
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5
votes
A problem about determinant and matrix
If there is a rational nonzero solution, there is an integer nonzero solution by multiplying up. At least one of the integers can be assumed odd by dividing out a common power of two.
The determinant …
- 37.7k
3
votes
When does $\det \begin{pmatrix} A & X \\ X^T & A \end{pmatrix} = (\det A)^2 + (\det X)^2$?
$$B=\left[\matrix{1&1&1&1\\1&1&-1&-1\\-1&-1&-1&1\\-1&1&-1&1}\right]$$
and probably many other solutions. I'm also voting to close because you didn't pose a research-level problem. If you have an int …
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2
votes
Accepted
Relation of row sums to largest eigenvalue
You are asking for relationships between the maximum independent set and the eigenvalues. If you search with those terms you will find several. For example, Haemers proved that the maximum size of a …
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1
vote
Adjacency matrix of total graph
If $C$ is the incidence matrix (rows indexed by vertices, columns by edges, two 1s in each column) then $CC^T$ is the Laplacian matrix and $C^TC-2I is the adjacency matrix of the linegraph. From this …
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3
votes
Accepted
Constant row-column sum matrices?
Since the vertices of the polytope are lattice points, the number of matrices is a polynomial of degree $(n-1)^2$ in $T$ for any fixed $n$ (the Ehrhart polynomial) but the polynomial is only known up to …
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2
votes
Accepted
Matrix completion problem with determinant condition?
I will prove it is NP-complete if $T$ is restricted to $\pm 1$.
Let $k_1,\ldots,k_n$ be an arbitrary list of integers.
Suppose the cofactors of $L$ along the top row are $c_1,\ldots,c_n$ and all not …
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