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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.
8
votes
Accepted
I have a very large sparse matrix, 'A', in Ax = b. What work in advance of getting 'b' can b...
Many have procedures for large sparse matrices. One method is "sparse $LU$ decomposition" (google that phrase for tutorials on it). …
- 37k
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".
- 37k
7
votes
Accepted
A partition of the set of all $n\times n\ (0,1)$-matrices
}$ is the number of matrices with row sums $r_1,\ldots,r_n$ and column sums $c_1,\ldots,c_n$. … This gives that the number of matrices with odd row sums and even column sums is $$2^{-2n}(2^{n^2}+(-1)^n 2^{n^2}).$$ Divide by the total $2^{n^2}$ to express it as a probability. …
- 37k
0
votes
Accepted
FInd smallest value $r$ such that a $n\times r$ matrix exists
There are plenty of ways to list all the $\lfloor r/2\rfloor$-subsets of $\lbrace 1,2,\ldots,r\rbrace$ such that adjacent members of the list have intersection $\lfloor r/2\rfloor-1$ (see "gray codes …
- 37k
14
votes
Accepted
Sum-regular $\{0,1\}$-matrices
I'm pretty sure this is unknown, though it would be great if I'm wrong.
Asymptotically there is a function $A(n)$ such that $M(n,n/2+t)\sim e^{-2t^2}A(n)$ for $t=o(n^{1/2})$, which follows from this …
- 37k
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 …
- 37k
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. …
- 37k
16
votes
1
answer
2k
views
Overlapping Gershgorin disks
What is a small counterexample for general matrices? Is there a counterexample for real symmetric matrices? Is there a nice family of matrices for which there is no counterexample? …
- 37k
2
votes
Computing the multiplicity of an eigenvalue of a 0-1 symmetric matrix...
The multiplicity of $\lambda$ is $n$ minus the rank of $\lambda I - A$, which you can find using Gaussian elimination. If you need to use floating-point arithmetic, you'll need to make decisions on w …
- 37k
3
votes
Invertibility of a certain matrix indexed by the Hamming cube
For any $i,j,k$, the automorphism group of $A$ is transitive on the set of pairs $(I,J)$ such that $|I|=i, |J|=j, |I\cap J|=k$. Therefore the same is true of the inverse (if it exists). That is, the …
- 37k
10
votes
Number of all different $n\times n$ matrices where sum of rows and columns is $3$
This problem was solved by Ron Read in his PhD thesis (University of London, 1958). Without requirement 2 there is a summation which isn't too horrible. With requirement 2 added as well, Read's soluti …
- 37k
8
votes
Solving the matrix equation $XX^t = A$ for binary matrix $X$
For general $A,X~$ this is a very difficult problem, but the condition you give that the rows of $X~$ have sum 2 makes it much easier. Consider each row to be an edge of a graph $G~$ (i.e. the two on …
- 37k
2
votes
Accepted
Number of symmetric matrix with fixed margins
We also give the exact values (the Ehrhart polynomial) for matrices up to order 9 and a conjecture for all row sums that has strong experimental backing. (Journal reference: J. Australian Math. …
- 37k
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 …
- 37k
1
vote
small sums of entries in submatrices - strange phenomenon
Unless I don't understand the question: Consider the submatrix formed by rows $R$ and columns $C$. Then the sum of this submatrix is
$$ \sum_{k\in R\cap C} x_k - \tfrac12 \biggl(\sum_{i\in R} x_i\b …
- 37k