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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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....
15 votes
3 answers
6k views

Questions on Toeplitz matrices: invertibility, determinant, positive-definiteness

These questions are probably very basic but I'll dare to ask them anyway since I didn't have much luck in Math Stack Exchange. Let $A$ be an $n \times n$ Hermitian Toeplitz matrix: $$A = \begin{...
ght's user avatar
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15 votes
1 answer
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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 ...
Vladimir Reshetnikov's user avatar
15 votes
1 answer
539 views

Matrix with small elements and prescribed determinant

Let $p$ be a large prime number. I want a $k\times k$ matrix with determinant $p$ and bounded integer elements (say, from -100 to 100). For which minimal $k$ such a matrix does always exist? We can ...
Fedor Petrov's user avatar
14 votes
2 answers
869 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. ...
T. Amdeberhan's user avatar
14 votes
1 answer
2k views

Why does this matrix have zero determinant?

This curious identity arose from studying reductions of the maximal ideal in certain monomial algebra. It can be proved "by hand", (i.e, using Macaulay 2), but I am seeking a more conceptual ...
Hailong Dao's user avatar
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14 votes
1 answer
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A determinantal formula

In my research, I encounter the following formula which I believe is correct (checked for $n\le3$). Is it classical ? If so, what is a reference ? I am given a real symmetric matrix $$S:=\int Y(t)Y(t)...
Denis Serre's user avatar
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14 votes
2 answers
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Are bounds known for the maximum determinant of a (0,1)-matrix of specified size and with a specifed number of 1s?

The problems of determining the maximum determinant of an $n \times n$ $(0,1)$-matrix and the spectral problem of determining exactly which other determinants can possibly occur are both reasonably ...
Gordon Royle's user avatar
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14 votes
1 answer
1k views

Expansion of $\det(A+B)$

If $A,B\in{\bf M}_n(k)$, then the following formula holds true: $$\det(A+B)=\sum_{r=0}^n\sum_{|I|=|J|=r}\epsilon(I,I^c)\epsilon(J,J^c)A\binom IJ B\binom{I^c}{J^c}.$$ In this formula, $I$ and $J$ are ...
Denis Serre's user avatar
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14 votes
1 answer
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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 ...
James Propp's user avatar
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14 votes
3 answers
822 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
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13 votes
3 answers
1k views

Is $-\det\big[\big(\frac{i^2+j^2}p\big)\big]_{1\le i,j\le (p-1)/2}$ always a square for each prime $p\equiv 3\pmod 4$?

Let $p$ be an odd prime and let $S_p$ denote the determinant $$\det\left[\left(\frac{i^2+j^2}p\right)\right]_{1\le i,j\le (p-1)/2}$$ with $(\frac{\cdot}p)$ the Legendre symbol. By Theorem 1.2 of my ...
Zhi-Wei Sun's user avatar
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13 votes
2 answers
869 views

How to invert the matrix [n choose 2j - i] ?

In a certain model of a stat-physics type, one encounters a matrix $$ A_n:=\left[\binom{n}{2j-i}\right]_{i,j=1}^{n-1}. $$ The determinant of this matrix (equal to $2^{\binom n2}$) counts the number ...
Leonid Petrov's user avatar
13 votes
2 answers
845 views

The expected square of the determinant of a random row stochastic matrix

In this question Anthony Quas asks about the expected absolute value of the determinant of an $n\times n$ row stochastic matrix $A$, where the rows are independently selected from the uniform ...
Richard Stanley's user avatar
13 votes
1 answer
391 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. ...
Will Sawin's user avatar
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13 votes
1 answer
376 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 ...
Wolfgang's user avatar
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13 votes
1 answer
594 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,\...
Marcel's user avatar
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13 votes
2 answers
421 views

Does $n^2$ divide $\det\left[\left(\frac{i^2+2ij+3j^2}n\right)\right]_{1\le i,j\le n-1}$ for each odd integer $n>3$?

For any odd integer $n>1$ and integers $c$ and $d$, define $$(c,d)_n:=\det\left[\left(\frac{i^2+cij+dj^2}n\right)\right]_{1\le i,j\le n-1},$$ where $(\frac{\cdot}n)$ is the Jacobi symbol. It is ...
Zhi-Wei Sun's user avatar
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13 votes
2 answers
1k views

Determinant of $V^* V$ where $V$ is rectangular Vandermonde matrix with nodes on unit circle

Let $z_{1},\dots,z_{k}$ be distinct complex numbers with $\left|z_{j}\right|=1,\;j=1,\dots,k$. For any natural $N\geqslant k$ consider the rectangular Vandermonde matrix $$ V_{N}=\begin{pmatrix}1 &...
dima's user avatar
  • 949
13 votes
0 answers
345 views

A determinant problem for primes $p\equiv 1\pmod4$

Let $p$ be an odd prime, and let $A_p$ denote the matrix $$[a_{ij}]_{1\le i,j\le (p-1)/2},$$ where $$a_{1j}=\left(\frac jp\right),\ \ \text{and}\ \ a_{ij}=\left(\frac{i^2+j^2}p\right)\ \text{for}\ i&...
Zhi-Wei Sun's user avatar
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12 votes
1 answer
401 views

Is the image of the map $A \to \bigwedge^k A$ closed over $\mathbb{R}$?

Let $V$ a real vector space of dimension $d$. Let $1<k < d-1$. Consider the map induced by the exterior algebra functor: $$ \psi:\text{End}(V) \to \text{End}(\bigwedge^kV) \, \, \, \, , \, \, ...
Asaf Shachar's user avatar
  • 6,611
12 votes
2 answers
2k views

Determinant of identity matrix plus Hilbert matrix

I am looking for the determinant $$ \det(I_n + H_n) $$ where $I_n$ is the $n \times n$ identity matrix and $H_n$ is the $n \times n$ Hilbert matrix, whose entries are given by $$ [H_n]_{ij} = \frac{...
Tobi's user avatar
  • 121
12 votes
2 answers
736 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\{ ...
user16215's user avatar
  • 830
12 votes
1 answer
618 views

Determinants: periodic entries $0,1,2,3$

Consider an $n\times n$ matrix $M_n$ where the sequence $$\{1,2,3,\dots,n^2\} \mod 4=\{1,2,3,0,1,2,3,\dots\}$$ forms a clock-wise spiral, in that given order. For example, $$M_4=\begin{bmatrix} 1&...
T. Amdeberhan's user avatar
12 votes
0 answers
457 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\...
Daniil Rudenko's user avatar
11 votes
3 answers
900 views

yet another determinant and inverse of a matrix

This problem is some variation of another MO question. Consider the matrix $$M_n:=\begin{bmatrix}-c& a & a& \dots & a \\ b & c & a& \ddots & a\\ b & b & -c &...
T. Amdeberhan's user avatar
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 $?
MPTuite's user avatar
  • 171
11 votes
1 answer
569 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. ...
T. Amdeberhan's user avatar
11 votes
1 answer
823 views

Pfaffian equals complex determinant?

Let $V$ be a Euclidean vector space and let $V^{\mathbb{C}} = V \oplus V$ be its complexification, with complex structure $$J = \begin{pmatrix} 0 & -\mathrm{id}\\ \mathrm{id} & 0 \end{pmatrix}....
Matthias Ludewig's user avatar
11 votes
1 answer
3k views

Exchange determinant and integral of a matrix-valued function

Assume $A(x)=(a_{ij}(x))_{k\times k}$ is a Hermitian matrix function on some manifold $M$, is there any inequality relates the integral of its determinant $\int_M \det(A)$ and the determinant of its ...
Joe's user avatar
  • 185
11 votes
2 answers
2k views

Determinant associated with a group action

Let $G$ be a finite group and $S$ be a finite set, with $G$ acting on $S$. I consider indeterminates $x_g$ indexed by $g\in G$ and form the matrix of the group action $A\in M_{S\times S}$. Its entries ...
Denis Serre's user avatar
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11 votes
1 answer
990 views

Do there exist nonzero identically vanishing polynomials over infinite (or characteristic zero) reduced indecomposable commutative rings?

Let $R$ be an infinite, characteristic zero, commutative ring. I can furthermore suppose it is reduced and indecomposable (no nontrivial nilpotents or idempotents). My question is whether there is a ...
Marcus Barão Camarão's user avatar
11 votes
2 answers
896 views

Determinant of arbitrary sum of positive semidefinite hermitian matrices

Suppose that $A_i$, $i=1,\ldots,m$ are positive semidefinite Hermitian $n\times n$ matrices, with $a_i^{(j)}$ being the $j$-th eigenvalue of $A_i$. Let $A_0=I$. QUESTION. Can we extend the result ...
VF1's user avatar
  • 253
11 votes
3 answers
1k views

A class of matrix determinants between Wronskians and Vandermondes

Update: see below Let $M$ be an $n\times n$ matrix that's constructed as follows. Construct the right-most column of $M$ as $[\alpha_1(x_1),\cdots,\alpha_n(x_n)]^T$ for some class of fixed functions $...
Alex R.'s user avatar
  • 4,902
11 votes
2 answers
946 views

How to prove this determinant is positive-II?

Question: Given an arbitrary number of real matrices of the form $ A_i= \biggl(\begin{matrix} C_i+E_i & B_i \\ B_i^T & D_i-F_i \end{matrix} \biggr) $, where $B_i$ is an arbitrary $n\times n$ ...
Lei Wang's user avatar
  • 845
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-...
Mare's user avatar
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11 votes
1 answer
609 views

Determinant of a certain Vandermonde matrix

Is there a closed form expression for the determinant of the $n\times n$ Vandermonde-type matrix $$A = \left(\begin{array}{} 1&g_1 & x_1&g_1 x_1 & x_1^2&g_1 x_1^2 & \cdots &...
Fred Hucht's user avatar
  • 2,705
11 votes
1 answer
713 views

Determinants of octonionic hermitian matrices

For quaternionic hermitian matrices (i.e. quaternionic square matrices $(a_{ij})$ satisfying $a_{ji}=\bar a_{ij}$) there is a nice notion of (Moore) determinant which can be defined as follows. ...
asv's user avatar
  • 21.1k
11 votes
0 answers
838 views

Is "Determinant" a Hochschild coboundary?

Assume that $n>2$. Is there an associative unital algebra structure on $\mathbb{C}^{n}$ such that $D$, the determinant as a $n-\text{form} $ on $\mathbb{C}^{n}$, would be a Hochschild ...
Ali Taghavi's user avatar
10 votes
2 answers
979 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 ...
T. Amdeberhan's user avatar
10 votes
2 answers
508 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 < \...
VSP's user avatar
  • 233
10 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?
Igor Demidov's user avatar
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) ...
inutard's user avatar
  • 103
10 votes
3 answers
14k views

Derivative of a determinant of a matrix field

Let $A(x_1,...,x_n)$ be an $n\times n$ matrix field over $R^n$. I am interested in the partial derivative determinant of $A$ in respect to $x_i$. In can be shown that: $\frac{\partial{\det(A)}}{\...
R S's user avatar
  • 985
10 votes
2 answers
3k views

Determinant of the "real part" of a matrix

Let $A$ be an $n\times n$ complex matrix, and write $A=X+iY$, where $X$ and $Y$ are real $n\times n$ matricies. Suppose that for every square submatrix $S$ of $A$, $|\mathrm{det}(S)|\leq 1$ (i.e., ...
Brian Street's user avatar
10 votes
4 answers
549 views

Hankel determinants of harmonic numbers

Let $H_n=\sum_{k=1}^n\frac 1 k$ be the $n$-th harmonic number with $H_0=0.$ Question: Is the following true? $$\det\left(H_{i+j}\right)_{i,j=0}^n=(-1)^n \frac{2H_{n}}{n! \prod_{j=1}^n \binom{2j}{j} \...
Johann Cigler's user avatar
10 votes
2 answers
1k views

When the determinant of a 2x2 polynomial matrix is a square?

Consider a 2x2 matrix $A$ with entries from $\mathbb{C}[x,y]$. Assume that $\mathrm{det} A$ is a square. Is it true that then $A$ can be represented as a noncomuting product $A=A_1 A_2 … A_{2n}$, in ...
mikhail skopenkov's user avatar
10 votes
2 answers
1k views

Is this a metric on the Grassmannian Manifold?

Let $m>n$ and consider the Set $$S_{m,n}=\{A \in \mathbb{R}^{m \times n}\lvert A^TA=I_n \}.$$ Does the function $d\colon S_{m,n} \times S_{m,n} \rightarrow \mathbb{R}$ defined by $$d(A,B)=\sqrt{1-\...
user35593's user avatar
  • 2,286
10 votes
1 answer
509 views

Homogeneous polynomials, mixed determinants, positive definiteness

Are there $n\times n$ real matrices $A_{1}, \ldots, A_{n}$ such that the $n$-homogeneous polynomial $$ f(x_{1}, \ldots, x_{n}) = \det(x_{1} A_{1}+\cdots +x_{n} A_{n}) $$ never vanishes on $\...
Paata Ivanishvili's user avatar
10 votes
1 answer
614 views

Is it true that $\det\big[\sin 2\pi\frac{(j-k)^2}p\big]_{1\le j,k\le p-1}=-\frac{p^{(p-1)/2}}{2^{p-1}}$ for each prime $p\equiv3\pmod4$?

Question. Does the equality $$\det\left[\sin 2\pi\frac{(j-k)^2}p\right]_{1\le j,k\le p-1}=-\frac{p^{(p-1)/2}}{2^{p-1}} $$ hold for every prime $p\equiv3\pmod4$? I have checked the equality numerically ...
Zhi-Wei Sun's user avatar
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