# Questions tagged [determinants]

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

475
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### 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(...

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51
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### 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)...

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32
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### Large sum of determinants of Hadamard products

We have $n$ by $n$ matrices $A$, $C$ and $S$ over a finite field $\mathbb{F}_q$. The $C$ is invertible of order $m$ as an element of $GL(\mathbb{F}_q,n)$.
Is there an algorithm, polynomial in $n$, ...

1
vote

2
answers

130
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### 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; ...

3
votes

1
answer

114
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### 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

163
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### 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[\...

5
votes

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287
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### 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
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3
answers

863
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### 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

147
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### 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

780
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### 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
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208
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### $\min(\det(\mathbf{A}))$ 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

62
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### 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{...

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0
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426
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### 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

482
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### 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

116
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### 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

268
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### 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}...

2
votes

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135
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### "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
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381
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### 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

906
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### 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 (...

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164
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### 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

338
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.
...

13
votes

2
answers

736
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### 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

269
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### 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
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0
answers

74
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### 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

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answers

126
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### 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 ...

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86
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### 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

458
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### 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
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149
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### 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-...

7
votes

2
answers

712
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### Determinant of matrix with Stirling numbers as elements

After noticing that the determinant of an $n \times n$ matrix $A_n$ with elements $a_{i,j}=i^j$, $1 \le i \le n$, $1 \le j \le n$, is the superfactorial (product of the first $n$ factorials), I wanted ...

2
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159
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### $\det(I+A) \leq 1$ with equality if and only if $A$ is nilpotent

Suppose we are given the following:
An $n \times n$ matrix $A$ (with rational entries).
$\operatorname{trace}(A)=0$
$B=A+I$ where $I$ is the $n \times n$ identity matrix.
Question. What conditions ...

0
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0
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44
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### Determinant of barycenter of a hyperbolic-matrix

Let $A \in \operatorname{GL}(d, \mathbb{R})$ be a hyperbolic matrix. I want to show that $$\det((1-\alpha)A+\alpha\operatorname{Id})\geq 1,$$
where $0<\alpha<1$.
Attempt:
In $\operatorname{SL}(2,...

15
votes

4
answers

1k
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### Possible values of the determinant for matrices with elements $\{1, 0, -1\}$

For matrices with elements $\{-1, 1\}$ it is known from here that the possible absolute values of determinants of $n \times n$, $n \leq 6$ matrices with entries $\{-1, 1\}$ are as follows:
...

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0
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85
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### Determinant Inequality with unitary matrix

I come up with the following conjecture while doing my research, which is a determinant inequality.
I have tried to run the MatLab simulation to verify its sanity. It seems that the inequality is true....

6
votes

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188
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### Lindström-Gessel-Viennot from properties of the $Alt^k$ functor?

Let $A$ be the directed adjacency matrix of an acyclic directed graph, with variables as its nonzero entries (one for each edge). The $(a,b)$ entry of the matrix $(I-A)^{-1}$ is the sum over all paths ...

5
votes

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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 $d_i(A)$ ...

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74
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### Number of roots of a Vandermonde like complex determinant

I am originally interested in the determinant
$$
\left|\begin{array}{cccc}\exp(i\lambda_1\cdot x_1) & \exp(i\lambda_2\cdot x_1) & ... & \exp(i\lambda_n\cdot x_1) \\\exp(i\lambda_1\cdot x_2)...

5
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1
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257
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### Log determinant of quadratic form

I am reading a paper Cook and Forzani - Likelihood-Based Sufficient Dimension Reduction where the author uses the following result from matrix analysis but does not explain why it is true nor provide ...

3
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0
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154
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### 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 ...

1
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0
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106
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### Determinants associated with Stern's diatomic sequence

Consider the so-called Stern's triangle (refer to these slides by R. Stanley), we denote here by $a_n(k)$. In an article Some linear recurrences motivated by Stern’s diatomic array, Stanley provided ...

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0
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121
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### Hankel determinants of the sequence $(\binom{n}{m})_{n\ge0}$ and related sequences

I posted (https://math.stackexchange.com/questions/4363151/generating-functions-of-hankel-determinants-related-to-hoggatt-triangles) this question on Mathematics StackExchange but have not received a ...

3
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170
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### Learning about determinantal varieties

In my research I recently stumbled upon a problem which involves trying to identify whether a given projective variety is determinantal or, even stronger, determinantal of a particular form. For ...

5
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### Hankel determinants for q-Catalan numbers where q is a root of unity?

Let ${C_n}(q)$ be the weight of the Dyck paths of semilength $n$ where the upsteps have weight $1$ and the downsteps which end on height $i$ have weight $q^i$.
They satisfy ${C_n}(q) = \sum\limits_{j ...

2
votes

2
answers

252
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### Determinants of minors occurring 'within' determinant of full matrix

$A= (a_{ij})$ is an $n\times n$ symmetric positive matrix. It induces a quadratic form $f(x):= x^tAx$ on $\mathbb{R}^n$. $D_m$ denotes the determinant of the top left $m\times m$ submatrix of $A$ (or ...

1
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0
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55
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### On perfect matchings on planar graphs - is there a linear time deterministic algorithm?

The slides here provide a way to get a pfaffian orientation from Minimum Spanning Tree.
MST can be found in linear time if graph is planar and weights are $1$ and the slides give a linear time ...

0
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0
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140
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### Determinant of chain complexes

Let $\mathcal{C}$ be the category of bounded cochain complexes of $R$-modules for a commutative ring $R$. I am trying to prove the following formula involving determinant $\text{Det}(F)$ of a map of ...

2
votes

1
answer

110
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### Monotonicity of the determinant of symmetric Toeplitz Matrices

For simplicity, i focus on a particular Toeplitz symmetric matrix, so let $A_{ij} = a^{|i-j|}$ for $i,j=1,\dots,n$ and $0<a<1$ be a Kac-Murdock-Szegő (KMS) matrix, e.g., for n=4
\begin{equation}
...

2
votes

0
answers

105
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### Can the absolute value of fixed sized minors be arbitrarily ordered?

In an $m \times n$ matrix $X$, there are exactly $\binom{m}{r}\binom{n}{r}$ minors of size $r \times r$. Is it always possible to construct a real matrix $X$ such that the absolute value of the ...

3
votes

0
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181
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### Combinatorial interpretation of a determinant

This is a continuation of Worpitzky-like identities?.
Let $ r_k(x)=\prod_{j=1}^k {(\frac{x+j}{j}})^{\min(j,k-j)}.$
As Sam Hopkins has remarked $r_k(x)$ is the number of plane partitions in a $ \...

1
vote

1
answer

230
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### Condition on the probabilities for the $J\times J$ matrix $[ \Pr(X=j \mid Y=k) ]$ to be invertible

$\DeclareMathOperator\Pr{P}\newcommand\cPr[2]{\Pr(#1 \mid #2)}$I have a $J \times J$ matrix:
$$
M:= \begin{bmatrix}
\cPr{X=1}{Y=1} & \cPr{X=2}{Y = 1} & \cdots & \cPr{X=J}{Y = 1} \\
\cPr{X=...

6
votes

0
answers

73
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### Rank of matrix coming from cobordism computations

In a computation of Pontryagin-numbers of certain manifolds (see the appendix of https://arxiv.org/pdf/2109.10306.pdf for more context) we came across the following elementary problem:
Consider the ...