Questions tagged [determinants]
Questions about the determinant of square matrices or linear endomorphisms. Also for closely related topics such as minors or regularized determinants.
500
questions
2
votes
0
answers
83
views
Statistics of perfect matching and incremental perfect matchings in bipartite planar graphs?
Planar graph permanent can be reduced to determinants and so statistics should be amenable.
Pick a uniformly random bipartite planar graph $G$ with $n$ vertices of each color and choose new additional ...
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0
votes
1
answer
209
views
Positive definite matrix
We have $a_1,a_2,...,a_n\in (0,1)$ and matrix
M=
\begin{bmatrix}2a_1&a_2&a_3&.&.\\a_2&2a_2&a_3&.&.\\a_3&a_3&2a_3&.&.\\.&.&.&.&.\end{...
- 163
9
votes
2
answers
420
views
Matrix of cosecants appearing in equivariant index computations
In a computation of characters of certain representations of finite cyclic groups which appear as equivariant indices of Dirac operators (using the Atiyah-Bott fixed point formula, cf. [1, Theorem 8....
- 186
8
votes
1
answer
611
views
Property of the trace on finitely generated projective modules
Let $A$ be a commutative ring with unit and let $P$ be a projective $A$-module finitely generated. By definition, there exists an $A$-module $P'$ such that $P\oplus P'$ is free of finite rank $r$. If $...
- 1,479
4
votes
0
answers
113
views
Positivity of q-analogs of central binomial coefficients?
With the usual $q-$notations
$[n]_q=1+q+\cdots+q^{n-1}=\frac{\,\,1-q^n}{1-q},$
$[n]_q!=[1]_q[2]_q\cdots[n]_q$ and
$\binom{n}k_q=\frac{[n]_q!}{[k]_q!\cdot[n-k]_q!}$
let
$$b(n,k,r,q)=\det\left(q^{r\...
- 5,572
4
votes
1
answer
1k
views
Determinant involving traceless unitary hermitian matrices
Let $S$ be the set of complex $N\times N$ matrices that are traceless, unitary and hermitian.
A friend asked me the following question, motivated by a problem in condensed matter physics:
Is it ...
- 1,417
5
votes
0
answers
190
views
On the determinants $\det\left[(i\pm j)\left(\frac{i\pm j}p\right)\right]_{1\le i,j\le(p-1)/2}$
Let $p$ be an odd prime and define
$$D_p^+:=\det\left[(i+j)\left(\frac{i+j}p\right)\right]_{1\le i,j\le(p-1)/2}$$
and $$D_p^{-}:=\det\left[(i-j)\left(\frac{i-j}p\right)\right]_{1\le i,j\le(p-1)/2},$$
...
- 14.5k
7
votes
1
answer
462
views
On a problem for determinants associated to Cartan matrices of certain algebras
This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested ...
- 26k
12
votes
0
answers
464
views
More mysterious properties of Gram matrix
This is another question related to the mysterious properties of the Gram matrix in dimension $4$. Here's the previous question.
The following fact could be extracted from 0402087:
For any $a_i\...
- 4,296
3
votes
1
answer
358
views
Determinant of an "almost cyclic" matrix
Let $n\geq 3$, let $Z$ be the matrix of the cyclic shift (the companion matrix of $X^n-1$), and for $\mathbf{d}\in \mathbb{C}^n$ let
$\operatorname{diag}(\mathbf{d})$ be the diagonal matrix with $\...
- 3,160
10
votes
0
answers
237
views
Generalized eigen property of a matrix
Given a $n \times n$ invertible matrix $A$, I am interested in the set
$$
\mathcal{S}(A) = \{ D \textrm{ diagonal matrix } \mid \det(D - A) = 0 \}.
$$
Thus, for all eigenvalues $\lambda_i$, we have $...
- 909
6
votes
0
answers
198
views
Conjecture for a certain Cauchy-type determinant
Given the Cauchy-like matrix
$$
\mathbf X_M(q) = \left[ \frac{2}{\pi} \frac{
\Gamma\!\left(m - \frac{1}{2} \right)\Gamma\!\left(n + \frac{1}{2} \right)
}{
\Gamma(m)\,\Gamma(n)
}
\frac{m-\frac{3}{4}}
{\...
- 2,705
1
vote
0
answers
146
views
On $\det\left[\frac1{i^2-ij+j^2}\right]_{1\le i,j\le p-1}$ and $\det\left[\frac1{i^2-ij+j^2}\right]_{1\le i,j\le (p-1)/2}$
QUESTION. Is my following conjecture true? If true, how to prove it?
Conjecture. Let $p$ be a prime with $p\equiv5\pmod 6$, and define the matrices $A_p$ and $B_p$ by
$$A_p:=\left[\frac1{i^2-ij+j^2}\...
- 14.5k
1
vote
0
answers
157
views
Non-trivial ways for generating matrices $A$ for which $A + A^T$ is positive-definite?
Disclaimer: This might be an SE question, but I'm not quite sure...
Thanks in advance!
Setup
So, it is known (see Proposition 5.2) that if $A + A^T$ is positive-definite then $A$ must be a $P$-...
- 6,726
11
votes
2
answers
551
views
Classification of algebras of finite global dimension via determinants of certain 0-1-matrices
I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background).
A Morita-...
- 26k
2
votes
2
answers
257
views
Equal-valued determinants in search of a proof: Part III
Encouraged by David's proof for my earlier MO question, let's consider a similar problem.
I can prove the below equality by computing each of the two sides, directly. That means, there is an ...
- 41.8k
11
votes
1
answer
570
views
Catalan determinants in search of a proof: Part II
This problem involves the Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$.
I can prove the below equality by computing each of the two sides, directly. That means, there is an algebraic proof.
...
- 41.8k
7
votes
1
answer
469
views
Deligne Pairing v.s. Weil Pairing on a Family of curves
We have the Deligne Pairing on a family of curve $\pi:X\to S$ by using
$$\langle L,M\rangle_{\mathrm{Pic}^0(X/S)}=\det R\pi_*(L\otimes M) \otimes (\det R\pi_*L)^{-1}\otimes (\det R\pi_*M)^{-1} \otimes ...
- 347
10
votes
2
answers
980
views
Determinantal symmetry: proof requested: Part I
Consider the determinantal function
$$F(a,b,c):=\det\left[\binom{i+j+a+b}{i+a}\right]_{i,j=0}^{c-1}.$$
I would like to ask:
QUESTION. Can you provide an argument, combinatorial or otherwise, to ...
- 41.8k
2
votes
1
answer
435
views
Coordinate free expression for the determinant of a $2 \times 2$ matrix
Let $E\rightarrow X$ be a rank two vector bundle on a variety $X$. How can one write an abstract map $Sym^2(E)\rightarrow (\wedge^2 E)^{\otimes 2}$ that in local coordinates, when we fix a frame of $E$...
user61586
5
votes
1
answer
94
views
On a maximum of a determinant with dependent variables
Let $x_1,\ldots,x_n\in [-1,1]^n$ and define the function
$$f(x_1,\ldots,x_n):= \prod_{i=1}^n\prod_{j=i}^n\left(1-\prod_{k=i}^j x_k\right).$$
This is a positive function, and actually coincides with ...
- 61
6
votes
0
answers
266
views
On the determinant $\det[\sec2\pi\frac{jk}p]_{0\le j,k\le(p-1)/2}$
On the basis of my computation, I have the following conjecture involving the secant function.
Conjecture. Let $p$ be an odd prime and define
$$S_p:=\det\left[\sec2\pi\frac{jk}p\right]_{0\le j,k\le (...
- 14.5k
6
votes
1
answer
505
views
A surprising identity: $\det[\cos\pi\frac{jk}n]_{1\le j,k\le n}=(-1)^{\lfloor\frac{n+1}2\rfloor}(n/2)^{(n-1)/2}$
On the basis of my computation, here I pose my following conjecture involving the cosine function.
Conjecture. For any positive integer $n$, we have the identity
$$\frac1{2n}\det\left[\cos\pi\frac{jk}...
- 14.5k
31
votes
1
answer
4k
views
Determinants of binary matrices
I was surprised to find that most $3\times 3$ matrices with entries in $\{0,1\}$ have determinant $0$ or $\pm 1$. There are only six out of 512 matrices with a different determinant (three with $2$ ...
- 12.3k
3
votes
0
answers
119
views
Does $(p-1)^2$ divide $\det[(\frac{i^2+cij+dj^2}p)]_{0\le i,j\le p-1}$ when $(\frac dp)=-1$?
Let $p$ be an odd prime. As in my paper, for $c,d\in\mathbb Z$ let us define
$$[c,d]_p:=\det\left[\left(\frac{i^2+cij+dj^2}p\right)\right]_{0\le i,j\le p-1},$$
where $(\frac{\cdot}p)$ is the Legendre ...
- 14.5k
2
votes
1
answer
212
views
On the determinant $\det[(\frac{i^2+dj^2}p)]_{0\le i,j\le(p-1)/2}$ with $(\frac dp)=-1$
Let $p$ be an odd prime. For $d\in\mathbb Z$ we define
$$T(d,p):=\det\left[\left(\frac{i^2+dj^2}p\right)\right]_{0\le i,j\le(p-1)/2},$$
where $(\frac{\cdot}p)$ is the Legendre symbol.
By (1.17) of my ...
- 14.5k
7
votes
2
answers
423
views
An upper estimate for $|\det(A+B)|$
If $A$ and $B$ are $n\times n$ matrices, then it easily follow from the definition of the determinant by sum over permutations, and from the Young inequality that
$$
|\det (A+B)|\leq C(n)(\Vert A\Vert^...
- 27.3k
4
votes
1
answer
473
views
A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (III)
As in Question 319254, for an odd prime $p$ and integers $c,d$ we define
$$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right),$$
where $(\frac{\cdot}p)$ is the Legendre symbol.
In my ...
- 14.5k
9
votes
2
answers
721
views
Certain matrices of interesting determinant
Let $M_n$ be the $n\times n$ matrix with entries
$$\binom{i}{2j}+\binom{j}{2i}, \qquad \text{for $1\leq i,j\leq n$}.$$
QUESTION. Is this true? There is some evidence. The determinant $\det(M_{2n+1})...
- 41.8k
2
votes
1
answer
185
views
On $\det[x+(\frac{i\pm j}p)]_{1\le i,j\le(p-1)/2}$ for primes $p\equiv 3\pmod 4$
I have made the followng conjecture on the basis of my computation.
Conjecture. For any prime $p\equiv3\pmod4$ with $p>3$, we have
$$\det\left[x+\left(\frac{i+j}p\right)\right]_{1\le i,j\le(p-1)/2}...
- 14.5k
8
votes
1
answer
1k
views
Determinant of "skew-symmetric" matrices
For $n\in\mathbb{N}$ and $m=\lfloor\frac{n}2\rfloor$, consider the $n\times n$ skew-symmetric matrix $A_n$ where each entry in the first $m$
sub-diagonals below the main diagonal is $1$ and each of ...
- 41.8k
6
votes
3
answers
1k
views
Expected determinant of random symmetric matrix with different Gaussian distributions of the diagonal and non-diagonal elements
Consider a random matrix $A \in \mathbb{R}^{N \times N}$ where the elements are random gaussian variables. The mean and variance of the elements are different on the diagonal and the off-diagonal:
$\...
- 355
24
votes
2
answers
3k
views
A Putnam problem with a twist
This question is motivated by one of the problem set from this year's Putnam Examination. That is,
Problem. Let $S_1, S_2, \dots, S_{2^n-1}$ be the nonempty subsets of $\{1,2,\dots,n\}$ in some ...
- 41.8k
4
votes
0
answers
99
views
Volume interpretation of number of perfect matchings in bipartite planar graphs
Permanent of biadjacency of bipartite graphs is the number of perfect matchings. In the case of planar graphs we can obtain an orientation with sign changes and get away with computing the determinant ...
- 13.7k
2
votes
0
answers
105
views
Understanding proof of q-theta function expression
In arXiv:math/0309252v4 at the bottom of page 11, the following result is proposed
$$ b_1 \theta(b_0 b_1^{\pm};p) \frac{z^{-1}\theta(z^2;p)}{\theta(b_0 z^{\pm};p)\theta(b_1 z^{\pm};p)} = (p;p)^{-2}\...
- 181
13
votes
1
answer
392
views
Is there a Giambelli identity with dual representations?
For natural numbers $a,b$ with $b\leq n-1$, let $V_{ (a|b)}$ be the irreducible representation of $GL_n$ with highest weight vector $(a+1, 1^b, 0^{n-b-1})$ where the exponentiation denotes repetition.
...
- 137k
7
votes
0
answers
138
views
A generalization of matrix minors to non-integer values
I am interested to know if there exist a notion of $k$-minors of a real square matrix, for non-integer positive values of $k$
One approach I thought of was to use the fact that the $k$-minors are (...
- 6,621
2
votes
1
answer
228
views
Does $\det[\lfloor(i^2+j^2)/p\rfloor]_{1\le i,j\le(p-1)/2}$ vanish for each prime $p>7$ with $p\equiv3\pmod4$?
Let $\lfloor x\rfloor$ be the floor function.
QUESTION: Does the determinant
$$D_p=\det\left[\left\lfloor\frac{i^2+j^2}p\right\rfloor\right]_{1\le i,j\le(p-1)/2}$$ vanish for each prime $p>7$ with ...
- 14.5k
7
votes
0
answers
389
views
Determinants associated to orthogonal polynomials
Let $${p_n}(x) = \sum\limits_{j = 0}^{n } {{{( - 1)}^{n - j}}p(n,j){x^j}} $$ be orthogonal polynomials satisfying $${p_n}(x) = (x - {s_{n - 1}}){p_{n - 1}}(x) - {t_{n - 2}}{p_{n - 2}}(x)$$ with ...
- 5,572
9
votes
3
answers
385
views
Determinant of a block matrix with many $-1$'s
For an array $(n_1,...,n_k)$ of non-negative integers and non-zero reals $a_1,...,a_k$, define a block matrix $M$ of size $n=n_1+\cdots+n_k$ as follows:
The main diagonal has blocks of sizes $n_i$ and ...
- 13.2k
15
votes
1
answer
467
views
Determinant of a matrix filled with elements of the Thue–Morse sequence
Let $n$ be a positive integer. Suppose we fill a square matrix $n\times n$ row-by-row with the first $n^2$ elements of the Thue–Morse sequence (with indexes from $0$ to $n^2-1$). Let $\mathcal D_n$ be ...
- 6,709
8
votes
1
answer
301
views
A symmetric bilinear form and a Plücker identity
It turns out that a special case of something I'm working on gives, as a corollary, a rather 19th-century-looking elementary statement about the rank of a certain symmetric matrix. I thought I would ...
- 584
5
votes
1
answer
2k
views
Is there a formula for the determinant of a block matrix of this kind?
I am looking for an expression that gives the determinant of a matrix of the form
\begin{bmatrix} A & B & 0 & \dots & 0 & C \\
B & A & B & & 0 & 0 \\
0 & ...
- 53
9
votes
1
answer
345
views
A binomial determinant formula: a new variant
In a previous MO question, the OP asks a proof for $\det_{1\leq i,j\leq n}\left(\binom{i}{2j}+\binom{-i}{2j}\right)=1$. Subsequently, Gjergji Zaimi generalized the problem to
$$\det_{1\le i,j\le n}\...
- 41.8k
7
votes
1
answer
244
views
The determinant of a $4\times4$ matrix associated to some specific polynomial as follow
Let $f\in \mathbb{R}[x_1,x_2,x_3,x_4]$ defined by
$$f_a(x_1,x_2,x_3,x_4)=\prod_{1\leqslant i<j\leqslant4}(x_i-x_j)^{2a_{ij}}$$
where $a=(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})\in \mathbb{N}^6$.
...
- 1,987
3
votes
1
answer
231
views
Reference request for some determinants of binomial coefficients
Let $C_{n}=\binom{2n}{n}\frac{1}{n+1}$ be a Catalan number. I am interested in books or papers where the following identities occur:
$$\det\left(\binom{i+j+1}{i-j+1}\right)_{0 \leq i,j\leq {n-1}}=C_{n}...
- 5,572
2
votes
0
answers
306
views
Determinant of a rank r perturbation
In the following paper:
Restricted Rank Modification of the Symmetric Eigenvalue Problem: Theoretical Considerations
on page 79, Golub et al. have the following set of equations:
$f(\lambda) = \...
- 121
4
votes
1
answer
283
views
integral kernel function for the SU(N) group
It is well know that the Haar probability measure for the $U(N)$ group, given by
$$
\begin{align}
dX_{U(N)} & = \frac{1}{N!(2\pi)^N}
\begin{vmatrix}
1 & 1 ...
- 333
9
votes
0
answers
340
views
Which polynomials in the minors of a matrix are invariant under conjugation?
$\newcommand{\Cof}{\operatorname{cof}}$
This is a cross-post.
Let $1<k<n$ be a fixed integer. I am trying to understand "what can be said" about the $k$-degree minors of a linear map $T:V \to ...
- 6,621
7
votes
3
answers
666
views
Distribution of sum of two permutation matrices
Determinant and permanent of sum of two $n\times n$ permutation matrices can be arbitrarily different.
What is the distribution of determinant of sum and difference of two $n\times n$ permutation ...
- 13.7k