Question about determinants, computation or theory. If $E$ is a vector space of dimension $d$, then we can compute the determinant of a $d$-uple $(v_1,\ldots,v_d)$ with respect to a basis.

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13
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457 views

Prove this determinant identity combinatorially

This is for those of you who understand the Lindstrom-Gessel-Viennot lemma. I am looking for a proof of the following identity using paths and such: Let $A$ be an $n\times n$ matrix, and for ...
7
votes
0answers
180 views

Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix. I have already asked a (viewed but unanswered) ...
7
votes
0answers
220 views

Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
7
votes
0answers
93 views

Maximum determinant of latin squares

I strongly conjecture that the maximum absolute determinant of a latin square can be attained by a circulant matrix. For example, $\pmatrix {5&4&2&3&1 \\ 1&5&4&2&3 \\ ...
7
votes
0answers
190 views

Determining sign(det(A)) for nearly-singular matrix A

Motivation: determining whether a point $p$ is above or below a plane $\pi$, which is defined by $d$ points, in a $d$-dimensional space, is equivalent to computing the sign of a determinant of a ...
6
votes
0answers
185 views

determinant of the Fubini-Study metric

Is there any easy way to compute the determinant of the Fubini-Study metric, given by: $g_{\alpha\bar{\beta}}=\frac{1}{1+\bar{z}z}\left(\delta_{\alpha\bar{\beta}}-\frac{\bar{z}_\alpha ...
6
votes
0answers
402 views

Determinant of Transpose of Linear Map

I'm trying to find a way to prove that the determinant of the transpose of an endomorphism is the determinant of the original linear map (i.e. det(A) = det(Aᵀ) in matrix language) using Dieudonne's ...
5
votes
0answers
71 views

calculation of the determinant of a block matrix little help

I need to prove $$\operatorname{det}\begin{pmatrix}A & B \\ C & D\\ \end{pmatrix}= \operatorname{det}(DA-CB),$$ where $A,B,C,D \in M_{n\times n}(R)$ with the property that $A$ and $B$ ...
5
votes
0answers
121 views

How to prove the determinant?

We have to prove the following result without expanding $\left|\begin{array}{lll} a^3 & a^2 &1 \\ b^3 & b^2 &1\\ c^3 & c^2 &1 \end{array} ...
5
votes
0answers
239 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
5
votes
0answers
503 views

Linearity of the determinant

I'd like to prove the following properties of the determinant map. $\det I = 1$ $\det$ is linear in the rows of the input matrix The determinant map is defined on $n\times n$ matrices $A$ by: ...
5
votes
0answers
336 views

Symmetric functions of the eigenvalues of A+B, A, B, ABA, BAB, et.c.

(this is an improved version of What about other symmetric functions of the eigenvalues? ) Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
5
votes
0answers
132 views

Determinant expression for the power sum

Let $S_{n,r} := \sum_{k=1}^{n} k^r$ be the power sum. On the homepage by W. Hecht (link) I have found the following determinant expression: $$S_{n,r} = (-1)^{r-1} \frac{n(n+1)}{(r+1)!} \det ...
4
votes
0answers
51 views

How can I determine the sign of a term in a determinant when the indices are out of order?

I am reading Shilov's book linear algebra. He explains how to compute determinants. Basically, for the plus terms you write \begin{equation} x_{a1}x_{b2}x_{c3}x_{d4}x_{e5} x_{f6} \end{equation} and ...
4
votes
0answers
45 views

Rank of a matrix whose all entries have the form $m^k$

The original problem is: Compute the determinant $$\begin{vmatrix} 1^k & 2^k & 3^k & \cdots & n^k \\ 2^k& 3^k & 4^k &\cdots & (n+1)^k \\ 3^k& 4^k ...
4
votes
0answers
69 views

Is the cone over Grassmannian manifold $Gr_2(\mathbb{C}^n)$ an open set of a determinantal variety?

Let $Gr_2(\mathbb{C}^n)$ the Grassmann manifold of the planes in $\mathbb{C}^n$. It is, via Plucker embedding, a projective variety. If we consider the cone $C$ over $Gr_2(\mathbb{C}^n)$, is it ...
4
votes
0answers
78 views

Expectation of the absolut value of the determinant of a random matrix

Let $A$ be a random matrix of size $m\times m$ with integer entries $-n\ldots n$. Each value should have the same probability. What is the expectation of the random variable $$X := |\det A|$$ Can ...
4
votes
0answers
33 views

Determinant of triangular matrix except for one column (atomic/Gauss/Frobenius)

Is there some "smart" way to calculate determinants that look like this? $\begin{vmatrix}-1&a_{1,2}&a_{1,3}&a_{1,4}&\cdots&a_{1,m-1}&a_{1,m} ...
4
votes
0answers
85 views

The determinant of a special matrix

Recently, I encounter the problem of calculating the determinant of the following matrix $$\left(\begin{array}{cccc} \sin(\theta_1) & \sin(\theta_1 + \delta_1) & \cdots & \sin(\theta_1 + ...
4
votes
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345 views

Understanding a proof about Hilbert Matrix

EDIT: I asked 3 questions. The first one I was able to solve myself, and the other two I cross-posted to MO. Lately I've been interested in the Hilbert Matrix (its definition will come later). I went ...
3
votes
0answers
55 views

Proving generalized Cassini's identity using determinant?

Motivation It is not hard to show, by using the general solution, that Proposition. If $(a_{n})_{n\in\Bbb{Z}}$ satisfies the recursive formula $ a_{n+2} = pa_{n+1} + qa_{n}$, then for any $n, i, ...
3
votes
0answers
40 views

Distinct $0,1$ symmetric circulant determinants

If $M$ is a circulant integer matrix of size $n\times n$ whose entries are randomly chosen from $\{0,1\}$ value, how many different determinants does $M$ possibly take value in? If $M$ is symmetric ...
3
votes
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49 views

Finding $n$ scalars such that $\det{(cI-A)}=0$ without eigenvalues

My problem is this Let $A$ be an $n\times n$ matrix over $\mathbb{F}$. Prove there are at most $n$ distinct scalars $c\in\mathbb{F}$ such that $\det{(cI-A)}=0.$ I know that the determinant is ...
3
votes
0answers
60 views

Prove that the determinant of a given matrix is proportional to the area of the triangle whose corners are the three points.

For three points in 2D, $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$, show that the determinant of \begin{bmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 2\\ x_3 & y_3 & 3\\ ...
3
votes
0answers
65 views

Probability that a random integer matrix is singular

Let $A$ be a $n\times n$-matrix with integers in the range $u..v$ , where $u<v$ are arbitary integers. Is there a formula, or at least, a good estimate, for the probability that the matrix is ...
3
votes
0answers
34 views

Determinant of a generalization of Moore matrices

The Moore matrix over $\mathbb{F}_q$ is the $n\times n$ matrix whose i'th row is: $a_i,a_i^q,a_i^{q^2},\dots,a_i^{q^{n-1}}$. The determinant of this matrix is the product of all linear combinations ...
3
votes
0answers
76 views

Invariants under a transformation

Consider a $j=1,\,SU(2)$ representation (or fundamental $SO(3)$ representation). Suppose that $a_1, b_i, c_i$ with $i=1,2,3$ are vectors transforming under this representation i.e ...
3
votes
0answers
52 views

How to prove that the determinant is the same no matter how you take it?

To find the determinant, pick a row and move along it creating minors and use the recursive definition of determinant. How do we know that the determinant will be the same no matter which row you ...
3
votes
0answers
155 views

Determinant proof

Let $A\in M_n(\mathbb C)$ and $\alpha \in \mathbb C$. If $B$ is the matrix obtained by multiplying a single row of $A$ by $\alpha$, then det$(B)=$ $\alpha$ det$(A)$. I'm trying to understand and use ...
3
votes
0answers
92 views

Prove that $\phi_1 \wedge \cdots \wedge \phi_k (v_1, \cdots, v_k) = \frac{1}{k!}\det[\phi_i(v_j)].$

I have proved these two exercises: (1) Suppose that $T \in \Lambda^p(V^*)$ and $v_1, \ldots, v_p \in V$ are linearly dependent. Prove that $T(v_1, \ldots, v_p) = 0$ for all $T \in \Lambda^p(V^*)$. ...
3
votes
0answers
148 views

How do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$?

The question is pretty straight-forward: how do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$ ?
3
votes
0answers
78 views

Identifying factors of higher order in a determinant

Consider a $n\times n$ matrix $A$ whose elements are some polynomials in the indeterminates $x_1, x_2,\ldots,x_m$. To calculate the determinant of such a matrix, one of the usual ways is to treat the ...
2
votes
0answers
36 views

Value of determinant using given conditions.

Let $A$ be a $2$ x $2$ matrix with real entries and $det(A)$ is equal to $d$ which is non-zero. It is given that $det(A +d(adjA))=0$ where $adj$ stands for the adjoint of the matrix. We have to find ...
2
votes
0answers
18 views

How to compute the following Jacobian

I need to show that the Jacobian of the n-dimensional spherical coordinates is $$\displaystyle r^{n-1}\sin^{n-2}\phi_1\sin^{n-3}\phi_2\cdots\sin\phi_{n-2}$$ then I have computed the Jacobian matrix, ...
2
votes
0answers
23 views

Determinants, Pfaffians, and…?

I recently stumbled across the wikipedia entry on Pfaffians and found them rather interesting, especially the property below. (assuming $A$ is a $2n\times 2n$ skew symmetric matrix) ...
2
votes
0answers
55 views

Is the determinant an analytic function?

I came accross a paper stating that the analytical property of determinants of complex matrices allows us to use some theorem for analytic functions. I am not able to confirm this since I am not sure ...
2
votes
0answers
38 views

Is the Cone over Grassmannian manifold a determinantal variety?

Let consider the Grassmann manifold $Gr(k,n)$ in the Plucker embedding and the Cone over $Gr(k,n)$, say $C(Gr(k,n))$. On the other hand consider $M$ the set of $n \times n$ skew-symmetric matrices. ...
2
votes
0answers
30 views

Log concavity/convexity of a determinant

I was wondering if anyone would be able to help me determine whether the following quantity is log concave or not with respect to $\alpha$? $$\left[\det(\textbf Y^\top \textbf P \textbf G \textbf ...
2
votes
0answers
52 views

Determinant of a sum

We have that: $\textbf Y \in \mathbb{R}^{n \times q}, \textbf G \in \mathbb{R}^{n \times n}, \textbf P \in \mathbb{R}^{n \times n}, \textbf Q \in \mathbb{R}^{q \times q}$. Furthermore, $\textbf G$ is ...
2
votes
0answers
29 views

Determining linear independence of three simple functions for a third order ODE. (2.9-7)

This is a very similar post to one previous by me but I felt that not all questions were satisfactorily answered. But I am sincerely grateful to those who tried. I would like a sharp independent eye ...
2
votes
0answers
52 views

Reference request on a sum-of-determinants identity

Suppose $X_1,X_2,X_3\in\mathbb R^{2\times1}$. Then $$ \det[ X_1,X_2] +\det[X_2,X_3] + \det[X_3,X_1] = \det[X_2-X_1,X_3-X_1]. $$ Where are this identity and higher-dimensional versions and their ...
2
votes
0answers
28 views

What is the simplest way to solve determinant of a $n \times n$ matrix by upper and lower triangular matrices?

I know the basic rules to solve for the determinant of an $n \times n$ matrix using upper and lower triangular matrices, but what is the simplest way?
2
votes
0answers
60 views

Weak convergence of determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
2
votes
0answers
45 views

Least number not being the determinant of a set of matrices

Let n > 1 be a natural number and u < v integers. How can I determine the least natural number not being the determinant of some n x n - matrix with integers in the range u..v without calculating ...
2
votes
0answers
73 views

Strange phenomena in determinants of matrix of determinants.

In my research, my computations are giving rise to the following strange phenomena: Let $$D=\begin{bmatrix}x_1^p & x_2^p & x_{3}^p\\ x_{1}^q & x_{2}^q & x_{3}^q\\ x_{1}^r & ...
2
votes
0answers
56 views

How to power series expand determinants?

Say $g$ is a ($d\times d$) matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same ...
2
votes
0answers
51 views

Differential Equations and Eigenvalues

I have the following system of differential equations: $$\left\{\begin{aligned} \frac {dx} {dt}=-4x+2y \\ \frac {dy} {dt}=-\frac 5 2x+2y \end{aligned} \right. $$ Which corresponds to the following ...
2
votes
0answers
138 views

Matrix determinant problem: Solution Verification

I previously posted Random determinant problem but did not understand the answer. Recently I came accross a solution method and would like to verify it here. Question: Is $$\mathbb ...
2
votes
0answers
64 views

Rank Of A Matrix Under Special Conditions

Let A be a $N*N$ matrix. Now A is defined in a special manner: Each row of A is defined by two integers L and R ($0\le L,R\le {N-1}$), such that all elements from the $L^{th}$ to the $R^{th}$ are all ...
2
votes
0answers
37 views

Determinant of Stirling submatrix

Consider the table containing the Stirling numbers of the second kind $S\left( n,k\right) $. From this table construct a square matrix $M$ containing $N$ differents rows from the table ...