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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12
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99 views

determinant of a standard magic square

What is the lowest positive, what the highest possible value for the determinant of a standard-magic-square-matrix of order n ? Are there singular standard-magic-square-matrices of any order ...
10
votes
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372 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
171 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
67 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 \\ ...
6
votes
0answers
98 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
168 views

Determinants of certain matrices.

I need help with this Linear Algebra homework problem; it's killing me. Problem 3: Consider the infinite dimensional matrix $A$ given by $A_{ij} = \left\{ \begin{array}{ll} 1, &\text{if } ...
5
votes
0answers
101 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
287 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
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241 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
120 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
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65 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
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116 views

What does abstract algebra have to say about the determinant?

The determinant is a homomorphism from the multiplicative monoid of matrices to the multiplicative monoid of a field (right?). I find this to be the most intuitive way to interpret some of the ...
4
votes
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69 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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269 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 ...
4
votes
0answers
351 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 ...
3
votes
0answers
24 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} ...
3
votes
0answers
26 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
68 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
49 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
115 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
81 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
119 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
76 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
42 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
52 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
42 views

Probability that a random integer matrix is singular

Let A be a nxn-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 singular ? ...
2
votes
0answers
50 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
41 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
96 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
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0answers
61 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
31 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 ...
2
votes
0answers
45 views

Reference request on pseudo-determinants

I am looking for a reference on pseudo-determinants$^{(1)}$. I am mostly interested on general and/or basic equalities and properties such as those obtained for determinants. Any pointers would be ...
2
votes
0answers
95 views

The most efficient algorithm to solve the following problem

Is there an efficient optimization algorithm to solve the following problem? $(\alpha,\beta,\gamma,\cdots) =$ argmax $\sum_{i}\log(\alpha a_i+\beta b_i+\gamma c_i+\cdots)$, s.t. ...
2
votes
0answers
86 views

Matrix inversion is to determinants as matrix logarithm is to what?

I have not put much effort into this question but I have thought about it for a year or so. Is there such thing as a "logarithmic determinant"? The starting point for this is that the determinant of ...
2
votes
0answers
91 views

Determinant, number of non zero columns

Trying to build a reduction from the maximum coverage problem to my research problem, I'm facing this difficulty : Let $X$ be a $n \times m$ binary matrix (with $m > n$), can we define a square ...
2
votes
0answers
169 views

Intuition in permutations for Laplace Determinant Expansion

Starting with the Leibniz formula for the determinant, I wish to derive the Laplace (Cofactor) Expansion. At the risk of being overly verbose, please see the proof here. Now I understand the idea of ...
2
votes
0answers
113 views

matrix construction

Given any matrix $A$, can one construct a matrix $B$ such that $B$ is nonnegative and the spectral radius of $B$ is strictly less than 1 the determinant of $A$ is equal to the first entry of $B^*$ ...
2
votes
0answers
114 views

Determinant of the Laplacian of a surface is this correct?

given a surface with metric $ g_{ab} $ i would like to evaluate the functional determinant of the Laplacian in the form $ - \partial _{s} \zeta (0,E^{2})=\log\det( \Delta + E^{2}) $ then i need to ...
2
votes
0answers
166 views

Determinants and homomorphisms of general linear groups

Consider the functions $\rho_1:M_1(\mathbb C)\to M_2(\mathbb R)$ where $$\rho_1(a+bi)=\begin{pmatrix} a&b\\ -b&a \end{pmatrix}$$ and $\rho_2:M_2(\mathbb C)\to M_4(\mathbb R)$ where ...
2
votes
0answers
272 views

Fastest integer matrix determinant software

I need to calculate vast numbers of determinants of integer matrices (size around 30x30 to 50x50) and would like to know the fastest software for this. It must use exact integer arithmetic as the ...
1
vote
0answers
24 views

Is finding a matrix out of some set with a given determinant a hard problem?

Given $n\ge 2\ \ ,\ u,v,k\ $ integers. Decision problem : Does a $n\times n$ - matrix with entries from $u$ to $v$ with determinant $k$ exist? In which complexity class is this problem ? Is it ...
1
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54 views

Teaching determinants

I am writing a first handout on determinants. The intended audience is confident with basic matrix algebra and the basic definitions of vector space theory. I just wondered if someone would comment on ...
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17 views

Further on determinants and finding the second partial derivative

Below is the question: $$\begin{cases} v+log\left|u\right|=xy \\ u+log\left|v\right|=x-y \end{cases}\implies \begin{cases} ...
1
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39 views

Minimum absolute determinant of a regular latin square matrix

It is easy to show that a latin square of size n x n has a determinant, which is a multiple of $\large \frac{n^2(n+1)}{2}$, if n is odd and $\large \frac{n^2(n+1)}{4}$, if n is even. This is a lower ...
1
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0answers
37 views

Complex matrices: looking for homomorphism

Let $\mathbb{C}$ denote the complex numbers, and let $M_2(\mathbb{R})$ be the ring of $2$ by $2$ matrices with real entries. Define a function $f:\mathbb{C} \to M_2(\mathbb{R})$ by $ f(a+bi) = ...
1
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0answers
47 views

probability of having a non-zero determinant

$K=\mathbb{Z}_p$ for some prime p, and $dim V = n$. It has been shown that the number of different bases in $V$ is: $\frac{1}{n!} \prod_{i=0}^{n-1}(p^n - p^i)$ (bases which are permutations of one ...
1
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0answers
43 views

Prove that every proper principal submatrix of $\lambda I-A$ is nonsingular under certain assumptions

Given that $A$ is a complex square matrix of order $n$, $\lambda$ is an eigenvalue of $A$ with geometric and algebraic multiplicity $1$, and $x,y$ are entrywise nonzero vectors such that $Ax=\lambda ...
1
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44 views

Calculate determinant of Vandermonde using specified steps.

$V_n(a_1,a_2\dots, a_n)$ is a $N\times N$ Vandermonde matrix = $$\left|\begin{array}[cccc] 11&z_1&\cdots&z^{n-1}_1\\ 1&z_2&\cdots&z^{n-1}_2\\ ...
1
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35 views

Cramer's rule and understanding Area/Volume

I'm having trouble connecting all the ideas we're learning in Linear Algebra. On the one hand, I understand how to find determinants, and therefore expansion factors. I also am fairly certain I have a ...
1
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53 views

Number of zeros of Wronskian

Is there some relation between the number of zeros of a Wronskian and properties of given functions? Having Wronskian (e.g. $2$ x $2$) $$W(x)=\left|\begin{array}{c}f_1(x) & f_2(x)\\f'_1(x) & ...