All Questions
11 questions with no upvoted or accepted answers
6
votes
0
answers
392
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Divisibility properties of minors of matrices
Let $A$ be an $m\times n$ matrix with integer entries. Let $d_i(A)$ be the greatest common divisor of all $i\times i$ minors of $A$, and define $d_0(A)=1$. Whenever $i\leq j$, one has that $d_i(A)$ ...
- 191
5
votes
0
answers
190
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Yet, another generalization of Catalan determinants
The discussion on this page is motivated by Johann Cigler's MO question. My intention arose from a possible generalization of Cigler's matrix
$$A_{n,m}=\left( \binom{2m}{j-i+m}-\binom{2m}{m-i-j-1} \...
- 43.2k
4
votes
0
answers
181
views
Fuss-Catalan: how does equality of these determinants hold?
There are many ways that the Catalan numbers seemed to have been generalized, one among them is through what Graham-Knuth-Patashnik (in Concrete Mathematics) dubbed as the Fuss-Catalan numbers
$\frac1{...
- 43.2k
3
votes
0
answers
185
views
"Circulant-Vandermonde" matrix: in search of a formula
An $n\times n$ circulant matrix $\mathbf{X}_n$ has the form
\begin{align}
\mathbf{X}_n= \begin{bmatrix}
x_1 & x_2 & \cdots & x_{n-1} & x_n \\
x_2 & x_3 & \cdots & x_n&...
- 43.2k
3
votes
0
answers
207
views
On a variation of the Vandermonde matrix
The ubiquitous Vandermonde matrix, of entries $(x_i^{j-1})_{i,j}^{1,n}$, and its determinant
$$\prod_{i<j}^{1,n}(x_j-x_i)$$
have found many utilities in Combinatorics and Physics, among other ...
- 43.2k
3
votes
0
answers
105
views
Can one do better than using general purpose determinant algorithms when using the Fisher-Kasteleyn-Temperley method for perfect matchings?
Questions.
(numerical.generalPfaffian) Is it proved anywhere that in general it is not easier0 to calculate the determinant (over $\mathbb{Q}$) of the skew-symmetric signed adjacency matrix defined ...
- 6,051
3
votes
0
answers
130
views
Where does this identity involving sums of Hankel-like determinants over partitions come from?
For a partition $\lambda=( \lambda_1,\dots,\lambda_n)\vdash n$ with $\lambda_1\ge\dots\ge\lambda_n\ge0$ and any function $f:\mathbb Z\to\mathbb C$, define a Hankel-like $n\times n$ matrix $$M_f(\...
- 13.4k
2
votes
0
answers
131
views
Pfaffian generalization
The identity
$$\left|
\begin{array}{cccc}
x & y_1 & y_2 & y_3 \\
z_1 & 0 & a & b \\
z_2 & -a & 0 & c \\
z_3 & -b & -c & 0 \\
\end{array}
\right|=\...
- 12.3k
1
vote
0
answers
159
views
Non-trivial ways for generating matrices $A$ for which $A + A^T$ is positive-definite?
Disclaimer: This might be an SE question, but I'm not quite sure...
Thanks in advance!
Setup
So, it is known (see Proposition 5.2) that if $A + A^T$ is positive-definite then $A$ must be a $P$-...
- 6,853
1
vote
0
answers
148
views
Traces in associative algebras
Are there some books or papers about the general definition of traces:
If $\mathscr{A}$ is an associative algebra over $K$ then the space of traces is the set of all linear functionals $\tau:\mathscr{...
- 283
0
votes
0
answers
283
views
A symmetric matrix with nonzero principal minors is cogredient to a diagonal matrix via an upper triangular
A paper I'm reading in representation theory states the following result:
Let $F$ be a field of characteristic zero, and $x$ a symmetric matrix in $M_n(F)$ all of whose principal minors are not zero. ...
- 6,180