Questions tagged [determinants]
Questions about the determinant of square matrices or linear endomorphisms. Also for closely related topics such as minors or regularized determinants.
28 questions from the last 365 days
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Special determinant formula
Consider two column vectors $\textbf{a}$ and $\textbf{b}$ of length $k$ and $m$ respectively, $km$ variables denoted $y_{i,j}$ (i=1 to k, j=1 to m), and a quadratic form $\textbf{y}^{T}\mathbb{M}\...
- 419
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168
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How to prove the following equation (which involves binomials and determinant of 2×2 matrices)?
I have tried many ways to prove the following equation, such as the method of induction and expanding all the terms in the summation,but things got more complicated.I could not find an appropriate ...
- 41
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1
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Determinant formula for a certain parametrized M-matrix
Let $P_{ij}$ be variables, and let $A \in \mathbb{R}^{n\times n}$ be the matrix defined by
$$
A_{ij} = \begin{cases}
-P_{ij} & i \neq j,\\
P_{i1} + P_{i2} + \dots + P_{in} & i=j.
\end{cases}
$$...
- 20.2k
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1
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154
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Determinant bundle over homogeneous varieties
I am looking for a way to compute the determinant of a homogeneous vector bundle over any homogeous variety. I am awere of how these computations work for the $A_n$ case (i.e., for flag varieties), ...
- 41
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51
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Degree of determinant of a (non-monic) matrix polynomial
Let $n=2, 3, \dots$ and consider the matrix polynomial $L(\lambda)=\sum_{k=0}^{\ell}A_k\lambda^k$, where $A_k \in \mathbb{C}^{n\times n}$.
In the so-called monic case (or that can be made monic by ...
- 11
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1
answer
102
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Minimally change matrix with determinant 0
In the following matrix equation, all coefficients $a_{ij}>0$ and all $a_i>0$ and the column sums in the matrix $A$ are all 0
(e.g. $-a_{11}+a_{21}+a_{31}=0$, etc.).
This means that
the ...
2
votes
0
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108
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Largest prime determinant of a binary matrix
Given an integer $n$, I want to prove the existence of an $n\times n$ binary matrix (with 0,1 entries), whose determinant is a prime number. What is a lower bound on the largest determinant that I ...
- 3,549
2
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1
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144
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Determinant of finite group representation
Let $G$ be a finite group with $n$ elements. Let $\rho\colon G \to \operatorname{GL}(V)$ be a linear representation of $G$, where $V$ is a finite dimensional $\Bbb C$-vector space.
The action extends ...
- 3,432
1
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1
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54
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Eigendecomposition of a Gram matrix with specified columns
I originally posted this question on Math SE, but I am starting to think it is more suitable to post it here. After waiting about two weeks, I didn't get any activity. The original question is linked ...
- 111
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Are automorphisms of matrix algebras necessarily determinant preservers?
Is every automorphism $\phi : A \to A$ of a subalgebra $A \subseteq M_n$ necessarily a determinant preserver?
I would assume that the answer is no in general, but I'm unable to find an example (or any ...
- 261
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3
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When does $\det(\frac{A+A^T}{2})=\det(A)$ for positive-definite $\frac{A+A^T}{2}$?
Setup: Let $A$ be a real square matrix and assume its symmetric part $\frac{A+A^T}{2}$ is positive-definite. The inequality
$$
\det\left(\frac{A+A^T}{2}\right) \leq \lvert\det(A)\rvert
$$
is known as ...
- 145
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0
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49
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Possible determinants of 01-matrices with at most three 1s in each row, column
As a function of $n$, what is the set of possible determinants of $n \times n$ matrices whose elements are 0s and 1s and have at most three 1s in each row and column?
I really enjoyed the problem ...
- 429
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37
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Expected (log)volume of projection of a cube onto a random subspace
Suppose $A$ is a full-rank $d\times d$ dimensional matrix. Let $U \in R^{d\times k}$ be a projection to projection matrix onto a uniformly chosen sub-space of dimension $k$ (for example, they can be ...
- 187
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94
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A conjectured generalization of Oppenheim's inequality, inspired by Horn-Yang's theorem
In this post, $A$ and $B$ are hermitian $n \times n$ positive semidefinite matrices.
It is well known that if $A$ has rank $n$ and if $B$ has only positive entries on its diagonal, then the rank of ...
- 5,215
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195
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Conjectural values of some determinants involving Legendre symbols (II)
Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ denote the Legendre symbol. Motivated by the evaluation of the determinants
$$\det\left[\left(\frac{j+k}p\right)\right]_{1\le j,k\le(p-1)/2}\ \ \text{...
- 15.6k
2
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1
answer
345
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What's the explicit value of this determinant
Let $n\ge2$ be a positive integer, and let $b_1,\cdots,b_n, c_1,\cdots, c_n$ be variables.
Recently, I met the following determinant:
$$\det A=\left|\begin{array}{cccc}
1 & b_1+c_1 & b_1^2+c_1^...
- 87
4
votes
0
answers
238
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Conjectural values of some determinants involving Legendre symbols (I)
$\newcommand\Legendre{\genfrac(){}{}}$Let $p$ be an odd prime, and let $\Legendre\cdot p$ be the Legendre symbol. In 2003, Robin Chapman evaluated the determinants
$$\det\left[\Legendre{i+j}p\right]_{...
- 15.6k
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Inequality involving minors of an orthogonal matrix
Fix $n \geq 3$ and take any orthonormal vectors $x,y,z \in \mathbb{R}^n$. Let also $A \in M_n(\mathbb{R})$ be a symmetric matrix with positive entries ($A_{ij} = A_{ji} > 0$). Is the following ...
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110
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Bounded density for determinant of GOE
Let $M$ a random GOE matrix, i.e. $M=(M_{i,j})$ is a symmetric matrix and the $M_{i,j},i\leq j$ are independent centred Gaussien entries with variance 1, except on the diagonal where the variance is $...
- 1,352
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Eliminating nullity for enhanced non-singularity
If we have an
$n\times n$ matrix $A$ with entries either $0$ or $1$, where all diagonal entries are $0$ and the rank is $k<n$, can we reach full rank by changing exactly $n-k$ zero off-diagonal ...
- 4,058
4
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2
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203
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Results of invertibility of a matrix involving the Szego kernel
In the context of reproducing kernel Hilbert spaces, the Szego kernel is the function $k(z_i,z_j)=\frac{1}{1-z_j\overline{z_j}}$.
Given $2n$ points $\{z_1,\ldots,z_n\},\{w_1,\ldots,w_n\}\in\mathbb{D}\...
- 167
5
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2
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420
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Maximum determinant of binary matrices with special properties
Let $A$ be an $n$-by-$n$ binary matrix (all elements are in $\{0,1\}$). It is known that the determinant of $A$ is bounded by $O(n^{n/2} / 2^n)$. I am looking for tighter upper bounds for matrices ...
- 3,549
2
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1
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94
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Testing for equal characteristic polynomials using a single determinant calculation
Let $A_1,A_2$ be $n\times n$ symmetric matrices over $\{0,1\}$, and let $p_1, p_2$ be their respective characteristic polynomials over the rationals.
If $p_1 \ne p_2$, then there is some positive ...
- 37.7k
9
votes
3
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861
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A curious equation on determinant----linear algebra or algebraic geometry?
I recently find a curious and unexplainable(as seems to me) equation on determinant as follows.
$$3\begin{vmatrix}
a_1 & b_1 & c_1 & d_1 \\
a_2 & b_2 & c_2 & d_2 \\
...
- 357
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Does an instance of this generalisation of the determinant exist?
Let $n$ be composite, $d$ a divisor greater than $1$ and $m=n/d$. Does anybody know if there is a general mapping $T$ from $n×n$ matrices to $m×m$ matrices that preserves the determinant? Over a field ...
- 1,245
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1
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525
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What is the mathematician's definition of the determinant? [closed]
I am trying really hard to find a good definition of the determinant.
I have looked virtually every single resource online and everybody gives a different answer:
sum of cofactors or minors https://...
- 179
16
votes
1
answer
784
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The determinant as a differential operator
According to Gårding, the determinant is a hyperbolic polynomial over the space $\mathbf{Sym}_n$ of real symmetric $n\times n$ matrices. More precisely, it is hyperbolic in the direction of the ...
- 52.3k
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Conditions on symmetric $3 \times 3$ matrices to satisfy the convex equality for cofactor and determinant
Given any $3\times 3$ finite set of symmetric matrices $A_i$ and positive real $a_i$ such that $\sum_ia_i=1.$ Is there any equivalent condition to the existence of skew symmetric matrices $X_i$ such ...
