# Questions tagged [determinants]

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

375
questions

**6**

votes

**2**answers

223 views

### Characteristic polynomial of checker matrix

For every integer $n > 0$, let $C_n$ be the $4n \times 4n$ matrix having $1$'s in all positions $(i, j)$ such that $i - j$ is even, $3$'s in the two diagonals determined by $|i - j| = 2n + 1$, and $...

**5**

votes

**1**answer

270 views

### How to compute a more general version of Vandermonde / Cauchy double alternant determinant

Consider some variables $\{X_i\}_{1\le i \le n}$, $\{Y_i\}_{1\le i \le n}$, and $\{W_i\}_{1\le i \le n}$. Does anyone know how to compute the following determinant?
$$
\det ~ \left(\frac{W_j^{i-1}}{...

**16**

votes

**2**answers

986 views

### How to prove the determinant of a Hilbert-like matrix with parameter is non-zero

Consider some positive non-integer $\beta$ and a non-negative integer $p$. Does anyone have any idea how to show that the determinant of the following matrix is non-zero?
$$
\begin{pmatrix}
\frac{1}{\...

**1**

vote

**0**answers

202 views

### Algebraic relation given by a 3x3 determinant

I just encountered a very curious relation in an algebra. A bit simplified, I am working in a (particular) non-commutative algebra, with some relations.
One particular relation is the following:
For (...

**7**

votes

**1**answer

235 views

### Has vol. 3A of Cullis's “Matrices and Determinoids” been scanned and vol. 3B been archived?

This is a borderline question, but I'm going to risk posing it.
Cuthbert Edmund Cullis (1875?-1955?) was a somewhat obscure British mathematician whose opus magnum was a multi-volume treatise called ...

**6**

votes

**1**answer

95 views

### Minors of low rank skew-symmetric matrix

Let $A$ be an $n\times n$ skew-symmetric matrix of rank $r$.
Given subsets $X$ and $Y$ of row and column indices respectively, let $A_{X,Y}$ denote the submatrix of $A$ obtained by only keeping rows ...

**13**

votes

**1**answer

346 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 ...

**2**

votes

**0**answers

210 views

### Characteristic polynomials of some special matrices

This is related to question Matrix-valued periodic Fibonacci polynomials.
I want to find integer-valued matrices $x$ such that the Fibonacci polynomials $f_n(x)$, defined by the recursion $f_n(x)=...

**1**

vote

**0**answers

54 views

### Linear algebra - For symmetric matrix X $\in S^n$, prove the $a^T X a$ = $\det X \det(X_{n-1})$ , where $a_i$ = $(-1)^i M_{in} $ [closed]

Suppose we have a symmetric matrix X$\in S^n$, and $X_k$ denotes the submatrix consists of first $k$ rows and columns of X. If $\det X < 0$, but $\det X_1, ..., \det X_{n-1} > 0$. Let $a_i=(-1)^...

**11**

votes

**0**answers

280 views

### Determinantal identities for perfect complexes

Let $S$ be a noetherian scheme. Let $V,W$ be vector bundles on $S$. There is a canonical isomorphism of line bundles
$$
{\rm det}(V\otimes W)\cong{\rm det}(V)^{\otimes{\rm rk}(W)}\otimes{\rm det}(W)^{\...

**0**

votes

**1**answer

39 views

### Determinant of a block matrix with dissimilar elements [closed]

I am looking for an expression that convert calculation of given quadratic form into determinant of some block matrix. I see this in that form (that may be incorrect):
$x^T A x = \begin{vmatrix} x^T ...

**0**

votes

**0**answers

26 views

### Name for product of square-free content of invariant factors of a matrix

Let $M(\lambda)$ be a possibly non-square polynomial matrix (over $\mathbb{R}$ or $\mathbb{C}$ is sufficient for me, but could be more general). By standard theory, it can be put into Smith normal ...

**2**

votes

**0**answers

79 views

### Naive generalization of determinant from matrices to higher rank tensors

Recall that using the Levi-Cevita symbol the determinant can be written as
$$\operatorname{det} A=\frac{1}{n!}\epsilon_{i_1\dots i_n}\epsilon_{j_1\dots j_n}A_{i_1j_1}\dots A_{i_nj_n}$$
Some ...

**5**

votes

**0**answers

159 views

### Determinantal formula for plane partitions of shifted shape

For $\lambda = (\lambda_1,\ldots,\lambda_{\ell})$ a partition, a (weak) plane partition of shape $\lambda$ is a filling of the Young diagram of $\lambda$ with nonnegative integers such that entries ...

**1**

vote

**0**answers

115 views

### Is this determinant well-known?

Let $n, k$ be any positve integers. I'm wondering if the following determinant is well known
$$D_{n,k}= \begin{vmatrix}1^k& 2^k & 3^k&\cdots & n^k \\2^k & 3^k & 4^k &\...

**4**

votes

**1**answer

275 views

### How can one define a kind of “Determinant” on reduced group $C^*$ algebra?

Let $A$ be a unital $C^*$ algebra which is equipped with a faithfull trace $T$. In particular we may consider $A=C^*_{\text{red}} (G)$ for some discrete group $G$. We consider the following ...

**6**

votes

**2**answers

409 views

### How large a subset of $\mathbb{F}_q^d$ can determine all determinants?

Denote by $\mathbb{F}_q$ a finite field with $q$ elements. For $\mathcal{P}$ be arbitrary subset of $\mathbb{F}_q^d.$ We define the set
$$S:= \left\{ \det([x_1,x_2,\dots,x_d]): x_1,x_2,\dots,x_d \in ...

**2**

votes

**2**answers

174 views

### Cartan determinants of subsets

Let $n \geq 3$ be fixed.
We associate to every subset $S \subseteq \{1,...,n-1 \}$ a number, which we call Cartan determinant of $S$ (see the end of this post for a representation theoretic background)...

**0**

votes

**1**answer

141 views

### What are all the possibilities of $A$ s.t. $\det(A)=k$?

Suppose we have $A \in M_3(\Bbb N\cup\{0\})$ s.t. sum of the elements of each row is $k $ for some fixed $k\in \Bbb N\cup\{0\}$. What are all the possibilities of $A$ s.t. $\det(A)=k$?
We can start ...

**13**

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 ...

**4**

votes

**0**answers

73 views

### Are mixed discriminants and hyper-determinants the same thing?

Premise
Hello, I'm trying to learn more about hyperdeterminants, but as I'm sifting through the literature I'm finding more and more papers on Mixed discriminant and I find their definitions ...

**2**

votes

**0**answers

106 views

### Have these polynomials been studied? (Perhaps as generalizations of Schur polynomials in vector variables?)

For $n\geq 3$, let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb{Z}^2$ be a collection of points in the plane with integer coordinates $\mathbf{a}_i = (a_{1i},a_{2i})$ where each $a_{1i} > 0$. For ...

**0**

votes

**0**answers

48 views

### Any trace-determinant inequalities for P/P0-matrices?

Two very purpolar inequalities involving trace and determinant are the following:
If $A$ is psd of size $n$, then $\frac{1}{n}\text{tr}(A) \ge \det (A)^{1/n}$.
If $A$ is psd, then $\log\det(A) \le \...

**2**

votes

**0**answers

28 views

### The total Wronskian

Given a sequence of function, $$F=\{f_1(x,t),f_2(x,t),f_3(x,t),\cdots,f_m(x,t)\},$$
we define the total Wronski determinant of this set of functions as
$$W(F)=\det\begin{vmatrix}F\\D_xF\\D_tF\\\vdots\\...

**1**

vote

**0**answers

83 views

### When does a matrix with high rank have a minor with disjoint rows and columns and high rank?

This is a somewhat open-ended followup question to
Does an antisymmetric matrix with high rank have a minor with disjoint rows and columns and high rank? and Does a non-singular matrix have a large ...

**1**

vote

**2**answers

72 views

### Determinant diagonal blocks compound matrix [closed]

Good afternoon,
I would like to prove the equation
\begin{equation}
\begin{vmatrix}
b_{1,1}I_d & b_{1,2}I_d & \cdots & b_{1,r}I_d \\
b_{2,1}I_d & b_{2,2}I_d & \cdots & b_{2,r}...

**9**

votes

**0**answers

98 views

### Cartan determinant of stable categories

Let $A$ be a finite dimensional algebra with finitely many indecomposable non-projective modules $M_1, M_2,...,M_n$.
Let $a_{i,j}:=\dim(\underline{Hom_A}(M_j,M_i))$ (the dimension of the stable Homs ...

**1**

vote

**0**answers

69 views

### Identifying a determinantal condition

Has the following condition already been studied and, if so, is there a known class of functions that satisfy it?
Condition. For a fixed $n > 0$, all the $2 \times 2$ minors of the matrix
$$
\...

**2**

votes

**0**answers

69 views

### Can we approximate this matrix field with an invertible matrix field?

Let $\mathbb{D}^2=\{ x \in \mathbb{R}^2 \, | \, |x| \le 1\}$ be the closed unit disk, and set
$$\begin{equation*}
A(x,y)=\left(
\begin{array}{cc}
x & -y \\
y & x
\end{array} \right)
\end{...

**10**

votes

**0**answers

245 views

### Interpretation of determinants on commutative rings

In real Euclidian space, the result of the determinant can be interpreted as the oriented volume of the image of the unit cube under an invertible linear map.
This interpretation conceptually depends ...

**2**

votes

**0**answers

49 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 ...

**0**

votes

**1**answer

137 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{...

**9**

votes

**2**answers

380 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....

**8**

votes

**1**answer

425 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 $...

**4**

votes

**0**answers

104 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\...

**3**

votes

**1**answer

308 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 ...

**5**

votes

**0**answers

176 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 $$D_p^{-}:=\det\left[(i-j)\left(\frac{i-j}p\right)\right]_{1\le i,j\le(p-1)/2},$$
...

**7**

votes

**1**answer

439 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 ...

**11**

votes

**0**answers

254 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\...

**3**

votes

**1**answer

208 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 $\...

**10**

votes

**0**answers

212 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 $...

**5**

votes

**0**answers

120 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}}
{\...

**1**

vote

**0**answers

126 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}\...

**2**

votes

**0**answers

134 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 $P$-...

**11**

votes

**2**answers

515 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-...

**2**

votes

**2**answers

227 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 ...

**11**

votes

**1**answer

457 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.
...

**7**

votes

**1**answer

275 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 ...

**11**

votes

**2**answers

895 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 ...

**3**

votes

**1**answer

385 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 $E$...