Questions tagged [determinants]
Questions about the determinant of square matrices or linear endomorphisms. Also for closely related topics such as minors or regularized determinants.
498
questions
7
votes
2
answers
556
views
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 \\
...
4
votes
0
answers
138
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 ...
0
votes
1
answer
387
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://...
16
votes
1
answer
707
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 ...
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 ...
5
votes
0
answers
231
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 ...
2
votes
0
answers
57
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 $\...
8
votes
3
answers
554
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]$$ to hold? Is ...
7
votes
0
answers
264
views
A conjecture about Hankel determinants of path generating functions
Let $a_{n,k}=a_{n,k}(x,c)$ be the generating function $\sum_P w(P),$ where $P$ runs over all paths from $(0,0)$ to $(n,k)$ consisting of horizontal steps $(1,0)$, up-steps $(1,1)$ and down-steps $(1,-...
9
votes
2
answers
433
views
Is there a determinantal point process proof of the Keating-Snaith formula for the cumulants of the log characteristic polynomial of a random matrix?
For $U$ a unitary $N \times N$ matrix, randomly distributed according to Haar measure, we have the complex-valued random variable $\log (\det (1-U))$. The real part and imaginary parts of $\log (\det (...
0
votes
0
answers
151
views
How to prove negativity of a $3\times3$ determinant whose elements involve trigamma, tetragamma, and pentagamma functions?
The classical Euler gamma function can be defined by the integral
\begin{equation*}
\Gamma(z)=\int_0^{\infty}t^{z-1}\operatorname{e}^{-t}\operatorname{d}t, \quad \Re(z)>0.
\end{equation*}
Its ...
0
votes
0
answers
60
views
Variance of the Barvinok estimator for Hafnians of nonnegative matrices
Let $A \in \mathbb{R}^{2m \times 2m}$ be a symmetric matrix with nonnegative entries. There is an approximate algorithm for estimating the Hafnian of such a matrix. Let $G$ be a skew-symmetrix matrix ...
2
votes
0
answers
37
views
Discrete approximation of continuous determinantal point processes
(throughout, "DPP" denotes "Determinantal Point Process")
TL;DR: Discrete DPPs are straightforward to compute with, continuous DPPs less so. Can we approximate continuous DPPs well ...
7
votes
0
answers
213
views
Why are these two determinants equal?
This question is a follow up on Mark Wildon's comment from an earlier MO question.
As usual, let $(q)_k=(1-q)(1-q^2)\cdots(1-q^k)$ with $(q)_0:=1$. Also, define the Gaussian polynomials by
$$\binom{n}...
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 ...
5
votes
0
answers
185
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
$$A_{n,m}=\left( \binom{2m}{j-i+m}-\binom{2m}{m-i-j-1} \...
1
vote
0
answers
100
views
On determinant and permanent of certain homotopy defined simple matrices
Let $A_1,A_2,B_1,B_2$ be four $n\times n$ $0/1$ square matrices where $$\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$$
hold ($per$ refers to permanent).
I. What ...
3
votes
1
answer
240
views
Determinant line of Fredholm operators and composition of morphisms
Let $P$ be a polarization of a Hilbert space $\mathcal{H}$, i.e. a bounded idempotent: consider a group $G=GL_{res}(\mathcal{H}):=\{g \in GL(\mathcal{H}): [g,P] \in HS\}$ (where $HS$ is the set of all ...
4
votes
0
answers
180
views
Fuss-Catalan: how does equality of these determinants hold?
There are many ways that the Catalan numbers seemed to have been generalized, one among them is through what Graham-Knuth-Patashnik (in Concrete Mathematics) dubbed as the Fuss-Catalan numbers
$\frac1{...
2
votes
0
answers
233
views
Determinants of band matrices which are related to Hankel matrices of Catalan numbers
Let $A_{n,m}$ be the band matrix $$ 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 &...
0
votes
0
answers
38
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}...
0
votes
0
answers
123
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_{...
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 $A\otimes A$...
0
votes
0
answers
104
views
I search representation in terms of Schur Q-function
Consider next sum
$$
Z_0^{N, N_f} = \sum_{r=0}^{N N_f} \sum_{\lambda \vdash r }s_{\lambda}(1^{N_f})
s_{\lambda} (1^{N_f})
= \det_{1\le i, j \le N} \ \binom{2 N_f}{N_f-i +j} = s_{N^{N_f}} \left(...
1
vote
0
answers
159
views
Generalized matrix determinant lemma for pseudo-determinant of symmetric matrix
The pseudo-determinant of a square matrix $A$ is the product of its nonzero eigenvalues. Consider the generalized matrix determinant lemma $$\det(A+UWV^\top) = \det A\det W\det(W^{-1} + V^\top A^{-1}U)...
1
vote
2
answers
146
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; ...
4
votes
1
answer
216
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 ...
2
votes
1
answer
199
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[\...
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 ...
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,...
3
votes
1
answer
464
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 ...
14
votes
3
answers
821
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.
\...
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 ...
2
votes
0
answers
117
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{...
0
votes
1
answer
764
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 ) \ \...
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.
$$
D = \begin{vmatrix}
0! & 1! & 2! & \ldots & x!\\
1! & ...
0
votes
1
answer
123
views
A variant of numeric Vandermonde which failed symbolically?
Given some variables $x_1, x_2, \dots, x_n$, the Vandermonde determinant is given by
$$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$ ...
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}...
3
votes
0
answers
168
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&...
5
votes
0
answers
414
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 &...
6
votes
2
answers
930
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 (...
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 ...
6
votes
1
answer
369
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.
...
15
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
288
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).$$
Let's ...
0
votes
0
answers
84
views
Finding the matrix for a given determinant
In my previous question asking about the co-intersection of three circles, a degree six polynomial in twelve variables was found for a special case. This polynomial has precisely 720 terms, of which ...
2
votes
0
answers
141
views
Applying 1D integral to matrix integral
In the proof for finding an analytic solution to the propagation of a Hermite-Gaussian beam though a paraxial system given in the paper "The elliptical Hermite–Gaussian beam and its propagation ...
0
votes
0
answers
96
views
Invertible matrices with bounded nonnegative coefficients
I am teaching a class in linear algebra and I asked myself the following question: what is the chance to get an invertible matrix if I write a random one? My impulsive answer is "very likely"...
2
votes
1
answer
485
views
Integer eigenvalues of a class of matrices inspired by Prof. Zhi-Wei Sun's conjecture
Theorem: Let $n>1$ be an odd number and $\zeta$ a primitive $n$-th root of unity. Then
\begin{eqnarray}
&&\sum_{\tau\in D(n-1)}\mathrm{sign}(\tau)\prod_{j=1}^{n-1}\frac{1}{1-\zeta^{j-\tau(j)...
4
votes
0
answers
156
views
An identity for Schur polynomials
Given a partition $\lambda$, the Schur polynomials can be defined, among many other ways, as
$$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-...