Questions tagged [determinants]

Questions about the determinant of square matrices or linear endomorphisms. Also for closely related topics such as minors or regularized determinants.

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A curious equation on determinant----linear algebra or algebraic geometry?

I recently find a curious and unexplainable(as seems to me) equation on determinant as follows. $$3\begin{vmatrix} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ ...
LichenSDU's user avatar
  • 143
4 votes
0 answers
137 views

Does an instance of this generalisation of the determinant exist?

Let $n$ be composite, $d$ a divisor greater than $1$ and $m=n/d$. Does anybody know if there is a general mapping $T$ from $n×n$ matrices to $m×m$ matrices that preserves the determinant? Over a field ...
Maarten Havinga's user avatar
0 votes
1 answer
369 views

What is the mathematician's definition of the determinant? [closed]

I am trying really hard to find a good definition of the determinant. I have looked virtually every single resource online and everybody gives a different answer: sum of cofactors or minors https://...
Olórin's user avatar
  • 179
16 votes
1 answer
683 views

The determinant as a differential operator

According to Gårding, the determinant is a hyperbolic polynomial over the space $\mathbf{Sym}_n$ of real symmetric $n\times n$ matrices. More precisely, it is hyperbolic in the direction of the ...
Denis Serre's user avatar
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0 votes
0 answers
46 views

Conditions on symmetric $3 \times 3$ matrices to satisfy the convex equality for cofactor and determinant

Given any $3\times 3$ finite set of symmetric matrices $A_i$ and positive real $a_i$ such that $\sum_ia_i=1.$ Is there any equivalent condition to the existence of skew symmetric matrices $X_i$ such ...
user519646's user avatar
5 votes
0 answers
227 views

$\log\det$ asymptotics of a skew-circulant matrix with additive diagonal bimodal disorder

I'd like to share a problem that I have been dealing with for a longer time now. In the framework of quenched disorder in the square-lattice Ising model I want to calculate, for large even $M$, the ...
Fred Hucht's user avatar
  • 2,625
2 votes
0 answers
55 views

Rank and determinant of the image of a vector bundle after applying a Schur functor?

Let $\mathcal{E}$ be a vector bundle of rank $r$ and degree $d$ over some smooth projective variety $X$. Furthermore, let $\lambda$ be a partition of $n$. We apply the $\lambda$-th Schur functor to $\...
Max Briest's user avatar
8 votes
3 answers
545 views

Jensen-like inequality for random matrix: $\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$

Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution. What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is ...
TheSimpliFire's user avatar
7 votes
0 answers
264 views

A conjecture about Hankel determinants of path generating functions

Let $a_{n,k}=a_{n,k}(x,c)$ be the generating function $\sum_P w(P),$ where $P$ runs over all paths from $(0,0)$ to $(n,k)$ consisting of horizontal steps $(1,0)$, up-steps $(1,1)$ and down-steps $(1,-...
Johann Cigler's user avatar
9 votes
2 answers
424 views

Is there a determinantal point process proof of the Keating-Snaith formula for the cumulants of the log characteristic polynomial of a random matrix?

For $U$ a unitary $N \times N$ matrix, randomly distributed according to Haar measure, we have the complex-valued random variable $\log (\det (1-U))$. The real part and imaginary parts of $\log (\det (...
Will Sawin's user avatar
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0 answers
150 views

How to prove negativity of a $3\times3$ determinant whose elements involve trigamma, tetragamma, and pentagamma functions?

The classical Euler gamma function can be defined by the integral \begin{equation*} \Gamma(z)=\int_0^{\infty}t^{z-1}\operatorname{e}^{-t}\operatorname{d}t, \quad \Re(z)>0. \end{equation*} Its ...
qifeng618's user avatar
  • 828
0 votes
0 answers
57 views

Variance of the Barvinok estimator for Hafnians of nonnegative matrices

Let $A \in \mathbb{R}^{2m \times 2m}$ be a symmetric matrix with nonnegative entries. There is an approximate algorithm for estimating the Hafnian of such a matrix. Let $G$ be a skew-symmetrix matrix ...
Alexey Uvarov's user avatar
2 votes
0 answers
36 views

Discrete approximation of continuous determinantal point processes

(throughout, "DPP" denotes "Determinantal Point Process") TL;DR: Discrete DPPs are straightforward to compute with, continuous DPPs less so. Can we approximate continuous DPPs well ...
πr8's user avatar
  • 688
6 votes
0 answers
210 views

Why are these two determinants equal?

This question is a follow up on Mark Wildon's comment from an earlier MO question. As usual, let $(q)_k=(1-q)(1-q^2)\cdots(1-q^k)$ with $(q)_0:=1$. Also, define the Gaussian polynomials by $$\binom{n}...
T. Amdeberhan's user avatar
0 votes
0 answers
55 views

Determinant of 2D non-positive second order partial differential operator

If I have an ordinary second order differential operator the Gelfand-Yaglom method is often useful to calculate its (regularized) determinant. The great advantage is that one doesn't have to calculate ...
Fetchinson0234's user avatar
5 votes
0 answers
184 views

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} \...
T. Amdeberhan's user avatar
1 vote
0 answers
100 views

On determinant and permanent of certain homotopy defined simple matrices

Let $A_1,A_2,B_1,B_2$ be four $n\times n$ $0/1$ square matrices where $$\det(A_1)=\det(A_2)=per(A_1)=per(A_2)=1$$ $$\det(B_1)=\det(B_2)=per(B_1)=per(B_2)=0$$ hold ($per$ refers to permanent). I. What ...
Turbo's user avatar
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3 votes
1 answer
235 views

Determinant line of Fredholm operators and composition of morphisms

Let $P$ be a polarization of a Hilbert space $\mathcal{H}$, i.e. a bounded idempotent: consider a group $G=GL_{res}(\mathcal{H}):=\{g \in GL(\mathcal{H}): [g,P] \in HS\}$ (where $HS$ is the set of all ...
truebaran's user avatar
  • 9,210
4 votes
0 answers
180 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{...
T. Amdeberhan's user avatar
2 votes
0 answers
233 views

Determinants of band matrices which are related to Hankel matrices of Catalan numbers

Let $A_{n,m}$ be the band matrix $$ A_{n,m}=\left( \binom{2m}{j-i+m}-\binom{2m}{m-i-j-1} \right)_{0\leq {i,j} \leq {n-1}}.$$ For example, $$A_{6,2}=\left ( \begin{matrix} 2 & 3 & 1& 0 &...
Johann Cigler's user avatar
0 votes
0 answers
36 views

Fischer Information and Entropy, matrix case, determinant of covariance matrix going to zero

If $X \sim \mathcal{N}(\mu, \sigma^2)$, then \begin{equation} \mathcal{I}\left(\mu, \sigma^2\right)=\left(\begin{array}{cc} \frac{1}{2\mathcal{H}...
Rémy Hosseinkhan Boucher's user avatar
0 votes
0 answers
122 views

Is it express in terms of Schur Q-function?

Consider next integral \begin{eqnarray} Z \ = \ h^{- N N_f} \ \int\limits_{SU(N)} \ dU \ \prod_{n=1}^{N} \ \det \left ( 1 + h U \right )^{ N_f} \ \left ( 1 + h U^{\dagger} \right )^{ N_f} \ = \sum_{...
Sergii Voloshyn's user avatar
5 votes
0 answers
158 views

Maximal minors of tensor product

Let $r \leq n$ be integers, and let $A$ be an $r \times n$ integer-valued matrix such that each $r\times r$ minor of $A$ is in $\{0, 1,-1\}$. Is it true that each $r^2 \times r^2$ minor of $A\otimes A$...
Ben's user avatar
  • 1,010
0 votes
0 answers
103 views

I search representation in terms of Schur Q-function

Consider next sum $$ Z_0^{N, N_f} = \sum_{r=0}^{N N_f} \sum_{\lambda \vdash r }s_{\lambda}(1^{N_f}) s_{\lambda} (1^{N_f}) = \det_{1\le i, j \le N} \ \binom{2 N_f}{N_f-i +j} = s_{N^{N_f}} \left(...
Sergii Voloshyn's user avatar
1 vote
0 answers
146 views

Generalized matrix determinant lemma for pseudo-determinant of symmetric matrix

The pseudo-determinant of a square matrix $A$ is the product of its nonzero eigenvalues. Consider the generalized matrix determinant lemma $$\det(A+UWV^\top) = \det A\det W\det(W^{-1} + V^\top A^{-1}U)...
Giacomo Petrillo's user avatar
1 vote
2 answers
144 views

Singularity of matrix pencil-like expression

I was working more on the topic on my previous question when I have to know whether the following statement is true to circumvent the "exception" caused by division by singular matrices; ...
Kanghun Kim's user avatar
4 votes
1 answer
186 views

A (bi)alternant formula for Wronskian

We know that there exists similarities between power functions and derivative of a function (in particular, Newton binomial formula and Leibniz rule for derivation of a product can be deduced from ...
Athena's user avatar
  • 275
2 votes
1 answer
198 views

A Vandermonde like determinant with exponentials

Let $n\geq m$ be non negative integers, and consider a list of $(n+m+1)$ distinct numbers (complex or real). I am interested in getting a closed form formula for the following determinant: $\det\left[\...
Athena's user avatar
  • 275
3 votes
0 answers
292 views

Decomposition of a determinant

Let $M$ be a $4\times 4$ symmetric matrix whose entries $m_{i,j}$ for $i,j =1,\dots,4$ are homogeneous polynomials of degree $2$ in $3$ variables. Assume that $m_{1,1} = 0$. Does there exist a ...
Puzzled's user avatar
  • 8,832
6 votes
3 answers
919 views

Proof of a matrix implication

If $A = \begin{bmatrix} x & 1\\ y & 0\end{bmatrix}, B = \begin{bmatrix} z & 1\\ w & 0\end{bmatrix}$, for $x,y,z,w \in \Bbb{R}$. I have observed by considering many examples of $x,y,z,...
BAYMAX's user avatar
  • 51
3 votes
1 answer
413 views

One question on block-circulant matrices

Circulant matrices are very useful in digital image processing. I found the general formula for determinant of circulant matrix. But I think it is not suitable for block-circulant matrices. For ...
user369335's user avatar
14 votes
3 answers
818 views

Determinant equal to Fibonacci sequence

I need to find the determinant of matrix defined by \begin{align*} & a_{i,1}=a_{1,j}=1,\quad \forall 1\leq i,j\leq n,\\ & a_{i,j}=a_{i-1,j}+a_{i,j-1}+i-j, \quad \forall 1< i,j\leq n. \...
Pascal's user avatar
  • 1,503
3 votes
2 answers
223 views

$\min(\det(\mathbf{A}))$ for special matrix $\mathbf{A}$

(The construction of matrix $\mathbf{A}$ is not difficult to be understood. You can first jump to A Toy Example to take a glance. Any idea or suggestion would be appealing for me.) The Original ...
BinChen's user avatar
  • 133
2 votes
0 answers
115 views

Jacobi formula for matrices: variations

Jacobi’s formula says: $\frac{d}{dt}\text{det}(A(t))=\text{det}(A(t)) \cdot \text{tr}(\text{Ad}(A(t))\cdot\frac{d}{dt}(A(t))$. Exists maybe a variation of the Jacobi’s formula where $\text{det}(\frac{...
Fynn13's user avatar
  • 83
0 votes
1 answer
759 views

How to calculate determinants of such types?

Consider next determinant that we want to expand around $h=1$ \begin{eqnarray} Z_q \ = \ h^{N N_f} \ \ \left ( \prod_{n=1}^{N} \ \sum_{l_n=0}^{N_f -q} \ h^{2l_n+q} \ \binom{N_f}{l_n} \right ) \ \...
Sergii Voloshyn's user avatar
4 votes
3 answers
532 views

Determinant with factorials is not 0?

Below is a simple determinant. I need to show that it is not 0, so that the corresponding matrix is invertible. $$ D = \begin{vmatrix} 0! & 1! & 2! & \ldots & x!\\ 1! & ...
sdd's user avatar
  • 109
0 votes
1 answer
123 views

A variant of numeric Vandermonde which failed symbolically?

Given some variables $x_1, x_2, \dots, x_n$, the Vandermonde determinant is given by $$V_n(x_1,\dots,x_n):=\det(x_j^{i-1})_{i,j=1}^n=\prod_{i<j}(x_j-x_i).$$ One can take as special cases: $x_j=j$ ...
T. Amdeberhan's user avatar
5 votes
1 answer
300 views

A conjectural permanent identity

Let $n>1$ be an integer, and let $\zeta$ be a primitive $n$th root of unity. By $(3.4)$ of arXiv:2206.02589, $1$ and those $n+1-2s\ (s=1,\ldots,n-1)$ are all the eigenvalues of the matrix $M=[m_{jk}...
Zhi-Wei Sun's user avatar
  • 14.4k
3 votes
0 answers
166 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&...
T. Amdeberhan's user avatar
5 votes
0 answers
412 views

Determinant of Hankel matrix with $a_n=(n!)^2$

Consider a Hankel matrix of the form $H_n(a_0(n))=\begin{pmatrix} a_0(n) & (1!)^2 & (2!)^2 & \cdots & (n!)^2\\ (1!)^2 & (2!)^2 & (3!)^2& \cdots & ((n+1)!)^2\\ (2!)^2 &...
fs98's user avatar
  • 51
6 votes
2 answers
929 views

Is it impossible for determinants of these matrices to both be negative?

Suppose $A,B \in M_{n}(\Bbb{R})$ such that $A = \left[C_{1}\middle|\frac{I}{0\dots0}\right], B= \left[C_{2}\middle|\frac{I}{0\dots0}\right]$ , where $A$ and $B$ have different first columns (...
BAYMAX's user avatar
  • 51
1 vote
0 answers
173 views

Some $p$-adic congruences involving permutations

Motivated by my study of determinants and permanents, here I present several conjectures on $p$-adic congruences involving permutations. As usual, we let $S_n$ be the symmetric group consisting of all ...
Zhi-Wei Sun's user avatar
  • 14.4k
6 votes
1 answer
368 views

Values of the determinants $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}\ (m=1,2,3,\ldots)$

For positive integers $m$ and $n$, let $D_m(n)$ denote the determinant $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}$, where the Kronecker delta $\delta_{jk}$ is $1$ or $0$ according as $j=k$ or not. ...
Zhi-Wei Sun's user avatar
  • 14.4k
14 votes
2 answers
1k views

Definitions of determinant by unique features

A well-known definition of the determinant is: The determinant is the only function of a vector space of dimension $n$ to its underlying field which is multilinear, alternating and normalized. See e....
3 votes
2 answers
286 views

Vandermonde $V_n$ mod $n$

Consider the all-familiar Vandermonde determinant $V_n(x_1,\dots,x_n)$ of the matrix of $(i,j)$-entries $M_n(i,j)=x_j^{i-1}$ so that $$V_n(x_1,\dots,x_n)=\prod_{1\leq i<j\leq n}(x_j-x_i).$$ Let's ...
T. Amdeberhan's user avatar
0 votes
0 answers
81 views

Finding the matrix for a given determinant

In my previous question asking about the co-intersection of three circles, a degree six polynomial in twelve variables was found for a special case. This polynomial has precisely 720 terms, of which ...
Thomas Blok's user avatar
2 votes
0 answers
141 views

Applying 1D integral to matrix integral

In the proof for finding an analytic solution to the propagation of a Hermite-Gaussian beam though a paraxial system given in the paper "The elliptical Hermite–Gaussian beam and its propagation ...
Alex's user avatar
  • 73
0 votes
0 answers
96 views

Invertible matrices with bounded nonnegative coefficients

I am teaching a class in linear algebra and I asked myself the following question: what is the chance to get an invertible matrix if I write a random one? My impulsive answer is "very likely"...
Andrea Marino's user avatar
2 votes
1 answer
482 views

Integer eigenvalues of a class of matrices inspired by Prof. Zhi-Wei Sun's conjecture

Theorem: Let $n>1$ be an odd number and $\zeta$ a primitive $n$-th root of unity. Then \begin{eqnarray} &&\sum_{\tau\in D(n-1)}\mathrm{sign}(\tau)\prod_{j=1}^{n-1}\frac{1}{1-\zeta^{j-\tau(j)...
Keqin Liu 'Kevin''s user avatar
4 votes
0 answers
156 views

An identity for Schur polynomials

Given a partition $\lambda$, the Schur polynomials can be defined, among many other ways, as $$S_{\lambda}(\xi_1,\dots,\xi_a)=\frac{\det\left(\xi_i^{\lambda_j+a-j}\right)_{i,j=1}^a}{\det\left(\xi_i^{a-...
T. Amdeberhan's user avatar

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