All Questions
Tagged with co.combinatorics determinants
95 questions
4
votes
0
answers
149
views
Generalization of a determinant with Lucas numbers and totient functions
Let $\gcd(a,b)$ denote the greatest common divisor function. H. J. S. Smith proved that
$$\det\left[\gcd(i,j)\right]_{i,j=1}^n=\prod_{k=1}^n\varphi(k),$$
where $\varphi(k)$ denotes Euler's totient ...
- 43.2k
19
votes
2
answers
576
views
Eigenvalues and eigenvectors of the matrix with entries $\dbinom{n+1}{2j-i}$ for $i, j = 1, 2, \ldots, n$
Let $n$ be a nonnegative integer, and let $B$ be the $n \times n$-matrix (over the rational numbers) whose $\left(i, j\right)$-th entry is $\dbinom{n+1}{2j-i}$ for all $i, j \in \left\{ 1, 2, \ldots, ...
- 33.8k
4
votes
1
answer
296
views
Is there a fast algorithm to test positivity of all principal minors of non-symmetric matrix?
I have a matrix $A \in \mathbb{R}^{n \times n}$ with positive eigenvalues. In the symmetric case, Sylvester's criterion implies that all the principal minors are positive. In the non-symmetric case, ...
- 188
10
votes
2
answers
537
views
Prove that the matrix $[\Gamma(\lambda_{i}+\mu_{j})]$ is nonsingular
Let A is a $n\times n$ matrix given by \begin{align*} a_{ij} = [\Gamma(\lambda_{i}+\mu_{j})] \end{align*} where $0 < \lambda_{1} < \ldots < \lambda_{n}$ and $0 < \mu_{1} < \ldots < \...
- 233
8
votes
0
answers
176
views
Nonzero subdeterminants conjecture: has anybody seen this anywhere?
I already posted this question on Mathematics StackExchange. A user there suggested that I rather post it on mathoverflow, since it is a research question. So here it is.
Let $m\geq2$, $n\geq1$ be ...
- 291
13
votes
1
answer
385
views
Some more binomial coefficient determinants
The setup is similar to this question, but generalizes the size of the Hankel matrix. We'll define
$$d(n,k,r):=\det\left(\binom{2i+2j+k+r}{i+j}\right)_{i,j=0}^{kn-1}.$$
Edit: Thanks to Johann ...
- 13.4k
13
votes
1
answer
626
views
A difficult determinant
(EDIT: I have removed the denominators I had in a previous version as they were superfluous)
The $N\times N$ determinant
$$D(a,\vec{b})=\det\left((2N+a+b_j-i-j)!\right)$$
has the nice form
$$D(a,\...
- 2,552
27
votes
2
answers
1k
views
Some binomial coefficient determinants
It is well known that for $n>0$
$$d(n)=\det\left(\binom{2i+2j+1}{i+j}\right)_{i,j=0}^{n-1}=1.$$
Computer experiments suggest that more generally
$$d(n,k)=\det\left(\binom{2i+2j+2k+1}{i+j}\right)_{i,...
- 5,622
8
votes
2
answers
1k
views
Does there exist a nonsingular graph for which the determinant of its adjacency matrix remains the same upon deleting a vertex?
Consider simple graphs. Any simple graph $G$ is called nonsingular if its $(0,1)$-adjacency matrix $A(G)$ has nonzero determinant. Does there exist any nonsingular simple graph whose determinant value ...
- 640
1
vote
0
answers
387
views
Relation between the sum of principal minors of different orders
Let $A$ be a symmetric (0,1)-square matrix of order $n$ having the diagonal entries zero. Let $m$ be the nullity of $A$ (number of zero eigenvalues), denoted by $\eta(A)$. Let $A_1$ be the square ...
- 640
1
vote
1
answer
96
views
Realising matrices as Cartan matrices
Given a matrix with natural numbers $\geq 0$ as entries and having determinant equal to one and positive diagonal entries. Is it the Cartan matrix of a finite dimensional algebra of finite global ...
- 26.5k
5
votes
2
answers
635
views
Some curious Hankel determinants
Let $f(n,q)=\prod_{j=1}^na(q^j)$ for a polynomial $a(q)$ and let $d(n)=\det(f(i+j,q))_{i,j=0}^n$ be its Hankel determinant.
Computer experiments suggest that
$$\lim_{q\to1}\frac{d(n)}{(q-1)^\binom{n+...
- 5,622
4
votes
2
answers
239
views
Distribution of $0$-$1$ matrices
Consider $n\times n$ matrices with entries in $\{0,1\}$. The determinants of these ranges from $0$ to the Hadamard bound $\frac{(n+1)^{\frac{n+1}2}}{2^n}$. Assume $n$ is large enough.
What does the ...
- 309
6
votes
0
answers
375
views
Monomial base change and the Vandermonde
Denote the falling factorials by $(x)_k=x(x-1)\cdots(x-k+1)$.
The Vandermonde determinant is given by $\det\left[x_i^{j-1}\right]_1^n=\prod_{i<j}(x_j-x_i)$.
It is well-known that in as much as ...
- 43.2k
14
votes
2
answers
873
views
"sinc'n determinant"
The function $\text{sinc}(x)=\frac{\sin x}x$ permeates mathematics and physics in several aspects, and it carries multiple presentations/formulations. My interest is to inject yet another one of such.
...
- 43.2k
8
votes
1
answer
321
views
"Almost Hankelized" numerical Vandermonde
One of the more utilized determinant is that of Vandermonde's
$$\begin{vmatrix}
1&x_1&x_1^2&\dots&x_1^{n-1}\\
1&x_2&x_2^2&\dots&x_2^{n-1}\\
\ldots&\ldots&\...
- 43.2k
4
votes
0
answers
96
views
Are extremal tournament matrices always circulant or 'almost circulant'?
Define an antisymmetric 1-x-matrix as an $n\times n$ matrix $M=(m_{ij})$ with $m_{ii}=0$ and $\{m_{ij},m_{ji}\}=\{1,x\}$ for all $1\le i<j\le n$. Call their set $\mathcal A_n$.
The setup is as ...
- 13.4k
3
votes
0
answers
184
views
Matrices with only two different entries and maximal determinant
Define $\mathcal M_n$ as the set of all $n\times n$ matrices of full rank with each entry either 1 or $x$.
I am interested in how big the determinant of such a matrix can be. For this, we define in a ...
- 13.4k
6
votes
0
answers
349
views
Exact determinant of a Cauchy-like matrix
Question. Is there a closed-form expression for the determinant of a $n \times n$ matrix $A$ with entries
$$
A_{i,j} = \frac{1 - \delta_{i, j}}{z_i - z_j}, \qquad 1\leq i, j\leq n,
$$
where $z_i$ ...
- 263
5
votes
2
answers
312
views
minimum-maximum entries matrix
Let $M(n)$ be an $n\times n$ matrix in the variables $x_1,\dots,x_n$ with entries
$$M_{i,j}(n)=\frac{x_{\max(i,j)}}{x_{\min(i,j)}}, \qquad 1\leq i,j\leq n.$$
I'm interested in the following:
...
- 43.2k
0
votes
0
answers
82
views
The effect of channel error on the determinant of transmitted matrix
Assume the following matrix
$$
E:=\left(
\begin{array}{ccccc}
e_1 & e_2 & \cdots & e_{p-1} & e_{p}\\
e_{p+1} & e_{p+2} & \cdots & e_{2p-1} & e_{2p} \\
\...
- 313
5
votes
1
answer
169
views
An extension of Hadamard maximum determinant problem
Consider the Vandermonde product $\prod_{1\le j < k \le n} |z_j - z_k|$. It is well-known that under the constraint $|z_j| \le 1$ for all $j$, the product is maximized at a picket fence ...
- 4,466
3
votes
1
answer
262
views
On a determinantal equality
In my study, I come across the following curious equality, which I do not know a proof yet (so I am asking it here).
Let $k$, $l\in \Bbb Z$ be fixed, $m$ --- the size of the below matrix $M$ --- is ...
- 305
3
votes
1
answer
385
views
Bounds for maximum determinant of circulant matrices
The Hadamard circulant conjecture states that there do not exist circulant Hadamard matrices with more than $4$ columns.
An $n$ by $n$ Hadamard matrix where the entries are chosen from $\{-1,1\}$ ...
- 3,377
9
votes
0
answers
188
views
Cycles of length $2^n - 2$ in the De Bruijn graph
It is well known that the number of (cyclic) De Bruijn sequences is $2^{2^{n-1}-n}$. This number may also be interpreted as the number of cycles of length $2^n$ in the De Bruijn graph of order $n$.
...
- 82.7k
8
votes
1
answer
406
views
Determinant of symmetric Latin square
Let $n=2m$ be an even number. Let us construct $n\times n$ symmetric matrices $S_n$ in the following way. The entries are indeterminates $X_1,\ldots,X_{n-1}$. We choose a $1$-factorization of the ...
- 52.3k
0
votes
0
answers
206
views
Finding a "special" non singular submatrix
Given a square integer matrix $A \in M_n(Z)$ and two subsets $I, J \subset \{ 1, \ldots, n\}$, we define $A_{I,J}$ as the sub-matrix of $A$ containing the rows (resp. columns) whose index is in $I$ (...
- 59
6
votes
1
answer
208
views
Sum identities with immanants
For $\chi$ being an irreducible character of the symmetric group $S_n$ and being $M$ a complex $n\times n$-matrix, I would like to show
$$
\sum_{\sigma, \rho \in S_n} \overline{\chi(\sigma)} \chi(...
- 93
12
votes
2
answers
780
views
Determinant of a checkerboard Hankel matrix with Catalan numbers
My goal is to compute
\begin{equation}
I = \det \left(\mathbf{I} + \mathbf{A}\right)
\end{equation}
where $\mathbf{A}$ is a $n \times n$ checkerboard matrix filled with Catalan numbers:
$$
\left\{
...
- 840
7
votes
5
answers
1k
views
How to calculate one Cauchy type determinant
As we know, a Cauchy determinant of size n admits the following explicit formula:
$$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x _j-x _i)(y _j-y _i)}{\...
- 113
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 $$M_f(\...
- 13.4k
6
votes
2
answers
1k
views
Determinant of the oriented adjacency matrix of a tree
Let $(V,E)$ be a finite oriented directed graph, with vertices and edges ordered, and $M$ the $|V|\times |E|$ matrix with entries
$$ m_{ve} = \begin{cases} 1 &\text{if $e$ points at $v$}\\
-1 &...
- 27.9k
7
votes
4
answers
1k
views
Generalized Cauchy-Binet sum over a fixed subset of indices
I originally posted this on math.stackexchange, but it quickly got buried. I removed it not too long after, thinking of rewriting it for MO, but I didn’t have a chance to post it until now. Apologies ...
- 13.6k
11
votes
2
answers
1k
views
A binomial determinant fomula
Is there an existing or elementary proof of the determinant identity
$
\det_{1\le i,j\le n}\left( \binom{i}{2j}+ \binom{-i}{2j}\right)=1
$?
- 171
4
votes
0
answers
657
views
determinant of fibonacci-sum graphs
We have a simple graph with vertices $\{v_1, v_2, ... v_n\}$.
The adjacency matrix of this graph is $A= (a_{ij})$ so that
$a_{ij}=1$ if $i+j$ belongs to the Fibonacci sequence;
$a_{ij}=0$ ...
- 41
8
votes
0
answers
488
views
det(A)det(B) = det(AB+correction), Capelli identities, "factorized" representation of $\mathfrak {gl}_n$
Context: Some probably know that there are Capelli identities which state
$$det(A)det(B) = det(AB+correction)$$ for some matrices with non-commuting elements, they go back to the 19-th century, but ...
- 24.9k
1
vote
0
answers
216
views
Generalized Schur polynomial from block Toeplitz matrices
By using the Jacobi-Trudi identity, one may interpret banded Toeplitz matrices, and minors of such matrices in terms of Schur polynomials, see for example
http://www-stat.stanford.edu/~cgates/PERSI/...
- 15.8k
7
votes
1
answer
578
views
To compute minors of Jacobian of symmetric polynomials
For any $n$ tuple $f_1,f_2,\dots,f_n$ in the polynomial ring $\mathbb{C}[x_1,x_2,\dots,x_n]$
one has Jacobian, expressed by the $(n \times n)$-determinants:
$$
J(f_1,\dots,f_n):=|\frac{\partial}{\...
- 446
14
votes
1
answer
1k
views
slick-proof-of-trick-for-counting-domino-tilings
The trick for rewriting the number of domino tilings of a simply-connected finite lattice region as the absolute value of the determinant of a matrix (due I believe to Kasteleyn and Percus, but if ...
- 19.7k
8
votes
0
answers
512
views
Can one give a "nice" expression for this determinant?
I am asking this question on behalf of a senior faculty member who is sometimes intimidated by computers. It is motivated by a problem in invariant theory. Unfortunately the question is a bit vague.
...
- 40.2k
2
votes
1
answer
295
views
A determinant involving only cyclotomic factors
Let $\alpha:\mathbb Z\longrightarrow \mathbb Z$ be a quadratic polynomial taking only integral values on the integers and consider the sequence of square-matrices with coefficients
$x^{\alpha(i+j)}$ ...
- 17.9k
11
votes
4
answers
5k
views
Maximum determinant of $\{0,1\}$-valued $n\times n$-matrices
What's the maximum determinant of $\{0,1\}$ matrices in $M(n,\mathbb{R})$?
If there's no exact formula what are the nearest upper and lower bounds do you know?
- 111
10
votes
2
answers
4k
views
Determinant of a $3\times3$ magic square
This is my first question with mathOverflow so I hope my etiquette is up to par here.
My question is regarding a $3\times3$ magic square constructed using the la Loubère method (see la Loubère method)
...
- 103
18
votes
3
answers
6k
views
Number of unique determinants for an NxN (0,1)-matrix
I'm interested in bounds for the number of unique determinants of NxN (0,1)-matrices. Obviously some of these matrices will be singular and therefore will trivially have zero determinant. While it ...
- 359
21
votes
2
answers
2k
views
Lifting matrices mod 2 to integers.
The following question was motivated by my research.
Consider a $n\times n$ matrix whose elements are $0$'s or $1$'s such that the determinant is odd. The question is: is it possible to assign signs ...
- 4,736