# Tagged Questions

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

**0**

votes

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133 views

### Is there an easy way to find the sign of this determinant without calculating it directly?

There exist real numbers $A_x, A_y, B_x, B_y, C_x, C_y, D_x$ and $D_y$. Is there an easy way to find the sign of following determinant without calculating it directly? BTW, this determinant appears ...

**8**

votes

**3**answers

183 views

### Determinant of a block matrix with many $-1$'s

For an array $(n_1,...,n_k)$ of non-negative integers and non-zero reals $a_1,...,a_k$, define a block matrix $M$ of size $n=n_1+\cdots+n_k$ as follows:
The main diagonal has blocks of sizes $n_i$ and ...

**7**

votes

**0**answers

128 views

+50

### 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 ...

**4**

votes

**1**answer

110 views

### A symmetric bilinear form and a Plücker identity

It turns out that a special case of something I'm working on gives, as a corollary, a rather 19th-century-looking elementary statement about the rank of a certain symmetric matrix. I thought I would ...

**3**

votes

**1**answer

225 views

### Is there a formula for the determinant of a block matrix of this kind?

I am looking for an expression that gives the determinant of a matrix of the form
\begin{bmatrix} A & B & 0 & \dots & 0 & C \\
B & A & B & & 0 & 0 \\
0 & ...

**6**

votes

**0**answers

158 views

### A binomial determinant formula: a new variant

In a previous MO question, the OP asks a proof for $\det_{1\leq i,j\leq n}\left(\binom{i}{2j}+\binom{-i}{2j}\right)=1$. Subsequently, Gjergji Zaimi generalized the problem to
$$\det_{1\le i,j\le n}\...

**7**

votes

**1**answer

174 views

### The determinant of a $4\times4$ matrix associated to some specific polynomial as follow

Let $f\in \mathbb{R}[x_1,x_2,x_3,x_4]$ defined by
$$f_a(x_1,x_2,x_3,x_4)=\prod_{1\leqslant i<j\leqslant4}(x_i-x_j)^{2a_{ij}}$$
where $a=(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})\in \mathbb{N}^6$.
...

**3**

votes

**1**answer

172 views

### Reference request for some determinants of binomial coefficients

Let $C_{n}=\binom{2n}{n}\frac{1}{n+1}$ be a Catalan number. I am interested in books or papers where the following identities occur:
$$\det\left(\binom{i+j+1}{i-j+1}\right)_{0 \leq i,j\leq {n-1}}=C_{n}...

**2**

votes

**0**answers

51 views

### Determinant of a rank r perturbation

In the following paper:
Restricted Rank Modification of the Symmetric Eigenvalue Problem: Theoretical Considerations
on page 79, Golub et al. have the following set of equations:
$f(\lambda) = \...

**0**

votes

**0**answers

19 views

### Determinants of block tridiagonal matrices when off-diagonal blocks are not m*m matrices

We know that if we have the block-tridiagonal matrix M(z) with blocks $A_i$, $B_i$ and $C_{i−1}$ (i =1,...,n) which are all m × m matrices
$$M(z)=\begin{bmatrix}
A_1 & B_1 & \cdots &\frac{...

**4**

votes

**1**answer

180 views

### integral kernel function for the SU(N) group

It is well know that the Haar probability measure for the $U(N)$ group, given by
$$
\begin{align}
dX_{U(N)} & = \frac{1}{N!(2\pi)^N}
\begin{vmatrix}
1 & 1 ...

**6**

votes

**0**answers

125 views

### Which polynomials in the minors of a matrix are invariant under conjugation?

$\newcommand{\Cof}{\operatorname{cof}}$
This is a cross-post.
Let $1<k<n$ be a fixed integer. I am trying to understand "what can be said" about the $k$-degree minors of a linear map $T:V \to ...

**7**

votes

**3**answers

261 views

### Distribution of sum of two permutation matrices

Determinant and permanent of sum of two $n\times n$ permutation matrices can be arbitrarily different.
What is the distribution of determinant of sum and difference of two $n\times n$ permutation ...

**7**

votes

**1**answer

234 views

### Block matrices and their determinants

For $n\in\Bbb{N}$, define three matrices $A_n(x,y), B_n$ and $M_n$ as follows:
(a) the $n\times n$ tridiagonal matrix $A_n(x,y)$ with main diagonal all $y$'s, superdiagonal all $x$'s and subdiagonal ...

**2**

votes

**0**answers

135 views

### Evaluate a curious determinant with Legendre symbol entries

Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. R. Chapman's conjecture on the exact value of the determinant of
$$C_p:=\left[\left(\frac{i-j}p\right)\right]_{0\le i,j\le (p-...

**7**

votes

**1**answer

383 views

### A new determinant question for primes $p\equiv3\pmod4$

Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ denote the Legendre symbol. Motivated by my question http://mathoverflow.net/questions/310301, here I introduce the matrices $A^+_p$ and $A^-_p$ ...

**12**

votes

**0**answers

252 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&...

**13**

votes

**3**answers

691 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 ...

**5**

votes

**0**answers

155 views

### Determinant arising in a problem from probability

Consider the determinant:
$$\Delta:=
\left|\begin{array}{cccc}
A_{j_1} & A_{k_1} & A_{j_1}A_{k_1} & 1 \\
A_{j_2} & A_{k_2} & A_{j_2}A_{k_2} & 1 \\
A_{j_3 } & A_{k_3 } &...

**1**

vote

**0**answers

59 views

### Determinant and restriction of scalar

Let $E/F$ be a finite separable extension of fields, and $V$ a finite dimensional vector space over $E$. Let $T\in\operatorname{End}_EV$ be a linear operator on $V$, and let $\det(T)$ be its ...

**16**

votes

**1**answer

543 views

### Lang's Jacobian identity: slicker, elementary proof?

In Jeffrey Lang, A Jacobian identity in positive characteristic, J. Commut. Algebra, Volume 7, Number 3 (2015), pp. 393--409, the following result is proven:
Theorem 1. Let $p$ be a prime. Let $\...

**3**

votes

**0**answers

279 views

### Evaluating a HUGE determinant

Fix a $k\times k$ positive definite symmetric real matrix $Y=(Y_{pq})$. Denote by $I_p$ the $p\times p$ identity matrix, $0_{p\times q}$ the $p\times q$ zero matrix, and $\delta_{pq}$ the Kronecker ...

**4**

votes

**0**answers

107 views

### Generalization of a determinant with Lucas numbers and totient functions

Let $\gcd(a,b)$ denote the greatest common divisor function. H. J. S. Smith proved that
$$\det\left[\gcd(i,j)\right]_{i,j=1}^n=\prod_{k=1}^n\varphi(k),$$
where $\varphi(k)$ denotes Euler's totient ...

**16**

votes

**2**answers

393 views

### Eigenvalues and eigenvectors of the matrix with entries $\dbinom{n+1}{2j-i}$ for $i, j = 1, 2, \ldots, n$

Let $n$ be a nonnegative integer, and let $B$ be the $n \times n$-matrix (over the rational numbers) whose $\left(i, j\right)$-th entry is $\dbinom{n+1}{2j-i}$ for all $i, j \in \left\{ 1, 2, \ldots, ...

**4**

votes

**1**answer

140 views

### Is there a fast algorithm to test positivity of all principal minors of non-symmetric matrix?

I have a matrix $A \in \mathbb{R}^{n \times n}$ with positive eigenvalues. In the symmetric case, Sylvester's criterion implies that all the principal minors are positive. In the non-symmetric case, ...

**10**

votes

**0**answers

419 views

### multi-dimensional integral of modified Vandermonde determinant

I'm looking for suggestions on how one might try to compute the following $(N-1)$-dimensional integral:
$$I_N= \frac{1}{(2\pi)^{N-1}(N-1)!} \int\cdots\int \\
\begin{vmatrix}
1 ...

**2**

votes

**0**answers

118 views

### lower bounding the absolute value of a determinant

In a problem that I'm working on currently, the following question came up and I feel this should be fairly elementary, but I couldn't prove it myself/couldn't find a reference. Any pointers or ...

**9**

votes

**2**answers

363 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 < \...

**1**

vote

**0**answers

92 views

### Decomposition of Determinant of Sub-Matrices of a Matrix

Consider an $n \times n$ matrix $\bf A$ over a field. Let $\bf A$ is constructed by the product of $n \times n$ matrices $B_i$, for $1\leq i \leq m$ which means
$$
{\bf A}=\prod_{i=1}^m\, {\bf B}_i\, ...

**8**

votes

**0**answers

356 views

### Determinant as a Hamiltonian

Are there two symplectic structures $\omega_1, \omega_2$ on $M_{2n}(\mathbb{R})$ such that the function $Det:M_{2n}(\mathbb{R})\to \mathbb{R}$ is completely integrable with respect to $\omega_{1}$...

**17**

votes

**2**answers

1k views

### How to prove positivity of determinant for these matrices?

Let $g(x) = e^x + e^{-x}$. For $x_1 < x_2 < \dots < x_n$ and $b_1 < b_2 < \dots < b_n$, I'd like to show that the determinant of the following matrix is positive, regardless of $n$:
...

**2**

votes

**0**answers

43 views

### Quasi-symmetric functions and determinants

In the field of symmetric functions, determinants show up all the time. For example, Jacobi-Trudi, Giambelli identity, definition of Schur polynomials as a quotient of determinants, and so on. This ...

**4**

votes

**1**answer

168 views

### Is the map $A \to \bigwedge^{k}A $ from matrices above rank $k$ proper?

$\newcommand{\End}{\operatorname{End}}$
Let $V$ be a $d$-dimensional real vector space. ($d \ge 3$). Fix an odd $2 \le k \le d-1$. Define
$H_{>k}=\{ A \in \End(V) \mid \operatorname{rank}(A) > ...

**14**

votes

**2**answers

300 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 ...

**5**

votes

**1**answer

133 views

### If $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$ equals $\bigwedge^k B$ for some complex matrix $B$, does it have a real source?

Let $1<k<d$ be an integer. Let $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$, and suppose that $A=\bigwedge^k B$ for some complex $B \in \text{End}(\mathbb{C}^d)$.
Does there exist $M \in \...

**9**

votes

**2**answers

320 views

### Does $\det\left[\left(\frac{i^2-\frac{p-1}2!\,j}p\right)\right]_{1\le i,j\le(p-1)/2}$ vanish for every prime $p\equiv3\pmod 4$?

For any odd prime $p$, let $D_p$ denote the determinant
$$\det\left[\left(\frac{i^2-\frac{p-1}2!\times j}p\right)\right]_{1\le i,j\le (p-1)/2},$$
where $(\frac{\cdot}p)$ is the Legendre symbol. Then
\...

**8**

votes

**2**answers

414 views

### Does $(2n)!$ divide $\det[(i^2+j^2)^n]_{0\le i,j\le n-1}$ for each integer $n>2$?

For $n=1,2,3,\ldots$ let $a_n$ denote the determinant $\det[(i^2+j^2)^n]_{0\le i,j\le n-1}$. Then
$$a_1=0,\ a_2=-1,\ a_3=-17280,\ a_4= 1168415539200.$$
QUESTION: Is it true that $(2n)!\mid a_n$ for ...

**4**

votes

**0**answers

164 views

### Does $n^2$ divide $\det[(i+j)^n]_{0\le i,j\le n-1}$ for each integer $n>2$?

For $n=1,2,3,\ldots$ let $a(n)$ denote the determinant $\det[(i+j)^n]_{0\le i,j\le n-1}$.
QUESTION: Is it true that $n^2\mid a(n)$ for all $n=3,4,\ldots$?
I even conjecture that
$$b(n)=\frac{(-1)^{...

**4**

votes

**1**answer

162 views

### Are the determinants $\det[\mu(i+j-1)]_{1\le i,j\le n}$ and $\det[\mu(i+j)^2]_{1\le i,j\le n}$ always nonzero?

Let $\mu$ be the well-known Mobius function.
QUESTION: Are the Hankel-type determinants
$$\det[\mu(i+j-1)]_{1\le i,j\le n}\ \ \text{and}\ \
\det[\mu(i+j)^2]_{1\le i,j\le n}$$
always nonzero?
It ...

**12**

votes

**1**answer

356 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) \, \, \, \, , \, \, ...

**8**

votes

**0**answers

141 views

### Nonzero subdeterminants conjecture: has anybody seen this anywhere?

I already posted this question on Mathematics StackExchange. A user there suggested that I rather post it on mathoverflow, since it is a research question. So here it is.
Let $m\geq2$, $n\geq1$ be ...

**6**

votes

**2**answers

291 views

### Probability of a large random integer Matrix to have zero determinant

Suppose we have a matrix $A \in \{0,1\}^{n \times n}$ where
$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad 1-p\end{cases}$$
I would like to ...

**3**

votes

**1**answer

136 views

### Distribution of eigenvalues of a Wishart matrix

Is there a known expression for the eigenvalue distribution of a matrix of the form
$$\sum\limits_{i=1}^n k_ia_ia_i^T$$
where $a_i \in \mathcal{R}^m$, with $n > m$, $a_i \sim \mathcal{N}(0,\Sigma)...

**9**

votes

**2**answers

458 views

### Parallelepiped is defined by the volumes of its faces

Let $v_1,...,v_n\in \mathbb{R}^n$ be linearly independent. The parallelepiped defined by these vectors is $P(v_1,...,v_n)=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Observe that while the ...

**0**

votes

**1**answer

147 views

### Prove a special form of Schur polynomial identity [closed]

Let $A_k$ be the $n\times n$ matrix defined by
$$
A_k=\left[
\begin{array}{}
1 & x_1 & x_1^2 & \cdots & x_1^{n-2} & x_1^k \\
1 & x_2 & x_2^2 & \cdots & x_2^{n-...

**12**

votes

**1**answer

268 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 ...

**13**

votes

**1**answer

407 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,\...

**16**

votes

**5**answers

1k views

### Expected value of determinant of simple infinite random matrix

Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where
$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$
I would like to ...

**1**

vote

**0**answers

82 views

### Traces in associative algebras

Are there some books or papers about the general definition of traces:
If $\mathscr{A}$ is an associative algebra over $K$ then the space of traces is the set of all linear functionals $\tau:\mathscr{...

**26**

votes

**2**answers

1k views

### Some binomial coefficient determinants

It is well known that for $n>0$
$$d(n)=\det\left(\binom{2i+2j+1}{i+j}\right)_{i,j=0}^{n-1}=1.$$
Computer experiments suggest that more generally
$$d(n,k)=\det\left(\binom{2i+2j+2k+1}{i+j}\right)_{i,...