Questions tagged [determinants]
Questions about the determinant of square matrices or linear endomorphisms. Also for closely related topics such as minors or regularized determinants.
500
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Why were matrix determinants once such a big deal?
I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. ...
63
votes
7
answers
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How to prove this determinant is positive?
Given matrices
$$A_i= \biggl(\begin{matrix}
0 & B_i \\
B_i^T & 0
\end{matrix} \biggr)$$
where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove the following?
$$\det \big( I + e^...
- 845
56
votes
21
answers
17k
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Wonderful applications of the Vandermonde determinant
This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has ...
54
votes
0
answers
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What did Gelfand mean by suggesting to study "Heredity Principle" structures instead of categories?
Israel Gelfand wrote in his remarkable talk "Mathematics as an adequate language (a few remarks)", given at "The Unity of Mathematics" Conference in honor of his 90th birthday, the ...
- 1,228
50
votes
7
answers
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Determinant of sum of positive definite matrices
Say $A$ and $B$ are symmetric, positive definite matrices. I've proved that
$$\det(A+B) \ge \det(A) + \det(B)$$
in the case that $A$ and $B$ are two dimensional. Is this true in general for $n$-...
- 511
47
votes
5
answers
4k
views
Is the determinant equal to a determinant?
Let $\det_d = \det((x_{i,j})_{1 \leq i,j\leq d})$ be the determinant of a generic $d \times d$ matrix. Suppose $k \mid d$, $1 < k < d$. Can $\det_d$ be written as the determinant of a $k \times ...
- 6,197
39
votes
6
answers
6k
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Linear transformation that preserves the determinant
It seems "common knowledge" that the following holds:
Let $T$ be a linear transformation on $n\times n$ matrices with complex coefficients that preserves the determinant. Then there exists ...
- 501
36
votes
4
answers
3k
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What are the applications of immanants?
Definitions of determinant:
$\det(A) = \sum_{\sigma \in S_n}\operatorname{sgn} \sigma\prod_{i}a_{i, \sigma(i)}$
and permanent:
$\mathrm{per}(A) = \sum_{\sigma \in S_n}\prod_{i}a_{i, \sigma(i)}$
...
- 3,301
31
votes
1
answer
4k
views
Determinants of binary matrices
I was surprised to find that most $3\times 3$ matrices with entries in $\{0,1\}$ have determinant $0$ or $\pm 1$. There are only six out of 512 matrices with a different determinant (three with $2$ ...
- 12.3k
30
votes
2
answers
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How to make the Capelli's identity less mysterious?
The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity
To ...
- 21.1k
29
votes
1
answer
3k
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Is there an explicit formula for the hessian of "Determinant"?
Let $f: G= \mbox{GL}(n,\mathbb{R}) \to \mathbb{R}$ be the determinant function.
Then $\mbox{Hess} (f)$ is a two linear map on $M_{n}(\mathbb{R})\simeq T_{e}(G)$ where $e$ is the neutral element of $G$,...
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28
votes
6
answers
4k
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Expressing $-\operatorname{adj}(A)$ as a polynomial in $A$?
Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\operatorname{det}(A-xI) = p_0 + p_1x + \dots + p_n x^n$.
I ...
- 427
28
votes
4
answers
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Jacobi's equality between complementary minors of inverse matrices
What's a quick way to prove the following fact about minors of an invertible matrix $A$ and its inverse?
Let $A[I,J]$ denote the submatrix of an $n \times n$ matrix $A$ obtained by keeping only the ...
- 4,354
28
votes
2
answers
15k
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Determinants in Graph Theory
In graph theory, we work with adjacency matrices which define the connections between the vertices. These matrices have various linear-algebraic properties. For example, their trace can be calculated (...
- 381
27
votes
2
answers
1k
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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,...
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26
votes
1
answer
1k
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Finding the closest matrix to $\text{SO}_n$ with a given determinant
$\newcommand{\GLp}{\operatorname{GL}_n^+}$
$\newcommand{\SLs}{\operatorname{SL}^s}$
$\newcommand{\dist}{\operatorname{dist}}$
$\newcommand{\Sig}{\Sigma}$
$\newcommand{\id}{\text{Id}}$
$\newcommand{\...
- 6,621
24
votes
2
answers
3k
views
A Putnam problem with a twist
This question is motivated by one of the problem set from this year's Putnam Examination. That is,
Problem. Let $S_1, S_2, \dots, S_{2^n-1}$ be the nonempty subsets of $\{1,2,\dots,n\}$ in some ...
- 41.8k
23
votes
7
answers
15k
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Expected determinant of a random NxN matrix
What is the expected value of the determinant over the uniform distribution of all possible 1-0 NxN matrices? What does this expected value tend to as the matrix size N approaches infinity?
- 241
23
votes
4
answers
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What are Picard categories, where can I learn more about them, and why should I care to?
I have the category-theoretic background of the occasional stroll through MacLane's text, so excuse my ignorance in this regard. I was trying to learn all that I could on the subject of tensor ...
- 1,489
22
votes
2
answers
13k
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Infinite matrices and the concept of "determinant"
Suppose we have an infinite matrix A = (aij) (i, j positive integers). What is the "right" definition of determinant of such a matrix? (Or does such a notion even exist?) Of course, I don't ...
- 1,861
22
votes
3
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On permanents and determinants of finite groups
$\DeclareMathOperator\perm{perm}$Let $G$ be a finite group. Define the determinant $\det(G)$ of $G$ as the determinant of the character table of $G$ over $\mathbb{C}$ and define the permanent $\perm(G)...
- 26.1k
22
votes
2
answers
1k
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Why is the catalecticant invariant under coordinate changes?
Let $\mathbf{k}$ be a commutative $\mathbb{Q}$-algebra. (We could play the
same game over any commutative ring $\mathbf{k}$, but this would be a bit more
technical, so let me avoid it.)
Fix a ...
- 33.2k
22
votes
3
answers
3k
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Splitting the determinant polynomial into linear factors - a Dedekind problem
Here's the question in a nutshell. For some $n\in\mathbb N$, we consider the polynomial
$\det\left(\left(X_{i,j}\right) _ {1\leq i\leq n,\ 1\leq j\leq n}\right)\in\mathbb Z\left[X_{i,j}\mid 1\leq i\...
- 33.2k
21
votes
2
answers
2k
views
Lifting matrices mod 2 to integers.
The following question was motivated by my research.
Consider a $n\times n$ matrix whose elements are $0$'s or $1$'s such that the determinant is odd. The question is: is it possible to assign signs ...
- 4,696
21
votes
1
answer
2k
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Vandermonde's remarkably clever notation for determinants
The entry on Alexandre-Théophile Vandermonde at the MacTutor History
of Mathematics archive ends with the description of the contents of Vandermonde's fourth and last mathematical paper, concluding ...
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20
votes
2
answers
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Connection between determinant and quotient rule
For the function $\dfrac{f(x)}{g(x)}$, we have, $\left(\dfrac{f}{g}\right)' = \dfrac{gf'-fg'} {g^2}$.
We can write the numerator as
$W(g,f) = \left|\begin{matrix} g & f \\ g' & f'\end{matrix}\...
- 385
20
votes
2
answers
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Formula expressing symmetric polynomials of eigenvalues as sum of determinants
The trace of a matrix is the sum of the eigenvalues and the determinant is the product of the eigenvalues. The fundamental theorem of symmetric polynomials says that we can write any symmetric ...
- 463
20
votes
2
answers
1k
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a determinantal identity
Dusan Pokorny and Jan Rataj have just posted a paper (http://arxiv.org/abs/1209.2305) in which they prove the identity
$$
\det (A-B) = \frac 1{d!} \sum_{k=0}^d (-1)^k \binom dk \det((d-k)A + kB)
$$
...
- 340
20
votes
1
answer
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When does the $4\times 4$ 'false Sarrus rule' compute the determinant correctly?
This question is most probably not research level, but I thought that the MO folks might like it... Feel free to close.
Here is the motivation: If you have ever taught a maths course for engineers ...
- 12.3k
19
votes
2
answers
2k
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 $n$:
...
- 325
19
votes
4
answers
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determinant of the table of characters
I am certain that the answer to this question exists somewhere. It might be a classical exercise.
Let $G$ be a finite group. Its table of characters is a square matrix, whose rows are indexed by the ...
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19
votes
3
answers
4k
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The Wronskian of sin(kx) and cos(kx), k=1...n
What is the determinant of the Wronskian of the functions $\{\cos\ x, \sin\ x, \cos\ 2x, \sin\ 2x,\ldots, \cos\ nx, \sin\ nx\}$? This determinant seems to be an integer, and the sequence starts with 1,...
19
votes
1
answer
933
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 $\...
- 33.2k
19
votes
2
answers
561
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, ...
- 33.2k
18
votes
3
answers
6k
views
Number of unique determinants for an NxN (0,1)-matrix
I'm interested in bounds for the number of unique determinants of NxN (0,1)-matrices. Obviously some of these matrices will be singular and therefore will trivially have zero determinant. While it ...
- 359
17
votes
4
answers
2k
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Possible values of the determinant for matrices with elements $\{1, 0, -1\}$
For matrices with elements $\{-1, 1\}$ it is known from here that the possible absolute values of determinants of $n \times n$, $n \leq 6$ matrices with entries $\{-1, 1\}$ are as follows:
...
- 519
17
votes
2
answers
2k
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Square root of the determinant line
Let $\Sigma$ be a compact Riemann surface equipped with a spin structure (a square root of $\Omega^1_\Sigma$, denoted $\Omega^{1/2}_\Sigma$).
Let $\Gamma(\Omega^1_\Sigma)$ be the space of holomorphic ...
- 42.5k
17
votes
2
answers
2k
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A determinant inequality
Notation: Suppose $\mathbf{A}$ and $\mathbf{B}$ are positive definite matrices in $\mathbb{R}^{n\times n}$ such that $\mathbf{A} \succeq \mathbf{B}$ (Loewner order). Let $\mathcal{S}(n,k)$ be the set ...
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17
votes
1
answer
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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
$$
{\rm det}(V\otimes W)\cong{\rm det}(V)^{\otimes{\rm rk}(W)}\otimes{\rm det}(W)^{\...
- 3,046
16
votes
2
answers
2k
views
Proof that block matrix has determinant $1$
The following real $2 \times 2$ matrix has determinant $1$:
$$\begin{pmatrix}
\sqrt{1+a^2} & a \\
a & \sqrt{1+a^2}
\end{pmatrix}$$
The natural generalisation of this to a real $2 \times 2$ ...
- 263
16
votes
5
answers
2k
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Expected value of determinant of simple infinite random matrix
Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where
$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$
I would like to ...
- 355
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votes
2
answers
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How to prove the determinant of a Hilbert-like matrix with parameter is non-zero
Consider some positive non-integer $\beta$ and a non-negative integer $p$. Does anyone have any idea how to show that the determinant of the following matrix is non-zero?
$$
\begin{pmatrix}
\frac{1}{\...
- 163
16
votes
4
answers
1k
views
Maximum of the Vandermonde determinant / minimum of the logarithmic energy
The problem is to find the asymptotics (as $n\to\infty$) of the maximum (say $M_n$) of the Vandermonde determinant
$$V_n:=\prod_{0\le i<j\le n-1}(a_j-a_i)
$$
over all $a_0,\dots,a_{n-1}$ such ...
- 118k
16
votes
1
answer
2k
views
The determinant of the sum of normal matrices
Given two normal matrices $A,B\in M_n({\mathbb C})$
whose respective spectra are $(\alpha_{1},\ldots,\alpha_{n})$ and
$(\beta_{1},\ldots,\beta_{n})$, is it true that $\det(A+B)$ belongs to
the ...
- 51.6k
16
votes
1
answer
723
views
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 ...
- 51.6k
16
votes
1
answer
842
views
Hankel determinants of binomial coefficients
For $\{h_{n}\}_{n=0}^{\infty}$ a real sequence, denote by $H_{n}$ the $n\times n$ Hankel matrix of the form
$$
H_{n}:=\begin{pmatrix}
h_{0} & h_{1} & \dots & h_{n-1}\\
h_{1} & ...
- 2,188
16
votes
0
answers
782
views
Determinant inequality involving Hermitian, positive definite matrices
Let $A,B,C\in M_{n}(\mathbb C)$ be Hermitian and positive-definite matrices such that $A+B+C=I_{n}$.
Show that
$$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$
This question has been ...
- 269
15
votes
3
answers
3k
views
Determinant of a $k \times k$ block matrix
Consider the $k \times k$ block matrix:
$$C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \vdots & \...
- 301
15
votes
1
answer
8k
views
On the determinant of a class symmetric matrices
Consider the matrix $2\times2$ symmetric matrix:
$$
A_2=\begin{pmatrix} 1 & a_1 \\ a_1 & 1\end{pmatrix}.
$$
It's clear that the restriction $|a_1|<1$ implies that $\det(A_2)>0$. Moreover,...
- 563
15
votes
3
answers
5k
views
How to show a certain determinant is non-zero
For any $n$ distinct points $x_1,x_2 , \ldots , x_n$ on the real line show that
the matrix $M$ where $M(i,j) = e^{\lambda_j x_i} $ has non-zero determinant
where $\lambda_1 \lt \lambda_2 \lt \ldots \...
- 251