Questions tagged [determinant]

Questions about determinants: their computation or their 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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27
votes
0answers
1k 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 $i,j\in\{...
26
votes
0answers
606 views

Determinant of a matrix that contains the first $n^2$ primes.

Let $n$ be an integer and $p_1,\ldots,p_{n^2}$ be the first prime numbers. Writing them down in a matrix $$ \left(\begin{matrix} p_1 & p_2 & \cdots & p_n \\ p_{n+1} & p_{n+2} & \...
22
votes
0answers
467 views

Can some proof that $\det(A) \ne 0$ be checked faster than matrix multiplication?

We can compute a determinant of an $n \times n$ matrix in $O(n^3)$ operations in several ways, for example by LU decomposition. It's also known (see, e.g., Wikipedia) that if we can multiply two $n \...
10
votes
1answer
385 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 ...
9
votes
0answers
181 views

Can we show that the determinant of this matrix is non-zero?

Consider the following symmetric matrix $M= \begin{bmatrix} f(x) & f(2x) & \dots & f(nx)\\ f(2x) & f(4x) & \dots & f(2nx)\\ \vdots & \vdots & \dots &...
9
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0answers
84 views

Can we recover all matrix minors from some of them?

Let $k,n$ be natural numbers, $1<k<n$. Suppose we have an "unknown" invertible $n \times n$ matrix $A$ over a field of characteristic zero. (we do not know the entries of $A$). Can we ...
8
votes
0answers
86 views

Determinant of a symmetric matrix with entries on diagonals

I am interested in the calculation of the determinant of the $N\times N$ symmetric matrix \begin{equation*} \mathbf B = \left(\begin{array}{*{20}c} 2 & & -1& &-1& &\\ & 2 &...
8
votes
0answers
2k views

Determinant of non-square Jacobian

Suppose I have a 3d solid in ${\bf R}^4$ which can be parametrized by the function $F:W\subset{\bf R}^3\rightarrow{\bf R}^4$. Now suppose I want to calculate the volume of this solid. Then naively I ...
8
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0answers
2k 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: $$\det ...
8
votes
1answer
101 views

Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?

Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: \text{SL}_n(\mathbb{Z}) \to \text{SL}_n(\mathbb{Z}/m\mathbb{Z})...
8
votes
1answer
824 views

Directional derivative of the determinant

Please help me find the mistake in my derivation: Let $f:M_{n,n}(\mathbb{R}) \to \mathbb{R}$ be the determinant function, $f(A)=det(A)$. Let $p_A(x)$ denote the charecteristic polynomial of $A$. ...
7
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0answers
375 views

Determinant of Matrix is Not Zero (combinatorial proof?)

Fix $n>k>1$. Define $\mathcal {A}(i,j)$ be the set of all sets $A\subset \{1,\ldots,n\}$ such that: $A$ has $k-1$ elements, $i\not\in A$ and also $j\not\in A$. Also, for $A$ in $\mathcal {A}(...
7
votes
1answer
223 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} \right|=(ab+bc+ca)\...
7
votes
0answers
270 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
1answer
175 views

Can we determine the determinant?

Could someone prove that this determinant is not zero? $$\left| \begin{array}{cccc} 1^n & 2^n & \cdots & (n+1)^n \\ 2^n & 3^n & \cdots & (n+2)^n \\ ...
7
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0answers
906 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 \...
6
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0answers
250 views

How to prove existence of a solution of this determinant equation?

Let $D\in\mathbb{R}^{n\times n}$ be a real diagonal matrix where $\sum_i D_{ii}<0$. Let also $R\in\mathbb{R}^{n\times n}$ and $L\in\mathbb{R}^{n\times n}$ be real (possibly) non-symmetric (...
6
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0answers
88 views

Has this logarithmic volume functional been studied?

$\newcommand{\M}{\mathcal{M}} \newcommand{\N}{\mathcal{N}} \newcommand{\VolM}{\text{Vol}_{\M}} \newcommand{\VolN}{\text{Vol}_{\N}}$ This question is mainly a reference request. Let $\M,\N$ be $d$-...
6
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0answers
143 views

Analogs of Cayley-Hamilton theorem for Pfaffian

The Pfaffian $\text{pf}$ is defined for a skew-symmetric matrix which is also a polynomial of matrix coefficients. One property for Pfaffian is that $\operatorname {pf} (A)^{2}=\det(A)$ holds for ...
6
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0answers
84 views

Determinant of a $2\times 2$ real matrix when an eigenvalue is given

Let $A$ be a real $2\times2$ matrix. If $5+3i$ is an eigenvalue of $A$, the $\det(A)$ a. equals $4$ b. equals $8$ c. equals $16$ d. cannot be determined from the given information $\mathbf{My\ ...
6
votes
0answers
172 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 $B$ ...
6
votes
0answers
637 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 z_{\bar{\beta}}}...
6
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0answers
136 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 + (...
6
votes
0answers
661 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 ...
6
votes
1answer
889 views

Determinant of hermitian matrix

Let $M=A+iB$ be a complex $n \times n$ Hermitian matrix. First of all we know that $$(\det M)^2=\det \begin{pmatrix} A & -B \\ B & A \end{pmatrix}.$$ Also $\det \begin{pmatrix} A & -B \\ B ...
5
votes
0answers
51 views

Let $T:\mathbb{R}_n\rightarrow\mathbb{R}_n$ the linear operator defined by $T(p(x))=p(x)+p'(x)$ Calculate $det(T)$

Let $T:\mathbb{R}_n\rightarrow\mathbb{R}_n$ the linear operator defined by $T(p(x))=p(x)+p'(x)$ Calculate $det(T)$ My work: Let $B=\{1,x,x^2,...,x^n\}$ a basis for $\mathbb{R}_n$. Then, $T(1)=1$ $...
5
votes
2answers
103 views

Equation involving a determinant of a $n\times n$ matrix

I would like to solve the following equation involving a determinant of an $n \times n $ matrix: $$ \begin{vmatrix} 2\cos \theta & -1 & 0 & \cdots & 0 \\ -1 & 2\cos \theta & -1 ...
5
votes
0answers
209 views

Construct a matrix $M$ from $A$ and $B$ such that $\det(M)=\det(A)-\det(B)$

Given two $n \times n$ symmetric matrices $A$ and $B$, is there a generic way to construct a larger block matrix $M$ such that $\det(M) = \det(A) - \det(B)$? A simple block expression is desired, in ...
5
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0answers
80 views

A conjecture concerning the irreducibility of characteristic polynomials of Arndt matrices

Letting $n \in \mathbb{N}$, let $M_{n}$ denote the $n \times n$ binary matrix with ones along the main antidiagonal and everywhere below the main antidiagonal and ones along the antidiagonal two ...
5
votes
0answers
450 views

Determinant of a function

I was thinking about matrices and then why arent there matrices with uncountable many values? (Probably this conecpt already exists for a very long time, but i don't know it) Assume there are ...
5
votes
0answers
80 views

Does the sum of weights in Kirchhoff’s construction equal the Gram determinant?

Background: An electrical network is modeled by a complex. Branch current distributions $\mathbf I\in C_1$ are represented by $1$-chains; branch voltage drop distributions $\mathbf V\in C^1$ are $1$-...
5
votes
0answers
85 views

A determinantal equality

Mark Kac wrote a paper about asymptotics of determinants whose main diagonal is taken from a function $f$, with $-1$ on the super and sub-diagonals. Specifically, $$ D_n = \begin{vmatrix} f(1/n) &...
5
votes
0answers
91 views

Over what rings is the Hefferonian determinant unique?

Fix an $n\in\mathbb{N}$ and a field $\mathbb{K}$. A lot of texts in linear algebra like to define the determinant function on $\operatorname{M}_n\left(\mathbb{K}\right)$ as the unique function $\...
5
votes
0answers
58 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 &...
5
votes
1answer
276 views

A challenge question in determinant of real matrices!

Suppose that $n\in \mathbb N -\{1\}$ and $a_{11},a_{12},\ldots,a_{nn}$ are $n^2$ distinct real numbers, prove that there is some enumeration of $a_{ij}$'s like $b_{ij}\ (i,j=1,2,\ldots,n)$ such that,...
5
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0answers
264 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 \begin{...
4
votes
0answers
87 views

Show $\mathrm{det}(M)$ is well defined.

One intuitive way to approach studying the determinant of a given matrix $M$ is to inspire its formal definition in the signed volume of applying the corresponding transformation to the unit $n$-cube. ...
4
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0answers
61 views

Maximal determinants of zero-one matrices

Is it possible to find the maximal determinant of matrices in $\{0,1\}^{n \times n}$? If so, what do matrices with maximal determinant look like? For example, when $n=3$, it's not hard to see that ...
4
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0answers
66 views

Abtract explanation for $\det(A+B) = ∑_{k=0}^n \langle \Lambda^k A, \Lambda^{n-k} B \rangle$

Let $A$ and $B$ be $n×n$ matrices. If we expand the determinant of $A+B$ as a sum over all permutations of $[\![1,n]\!]$ and all choices of whether the coefficient comes from $A$ or from $B$, this ...
4
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0answers
33 views

Matrices with positive permutation products

Let $A=(a_{ij})$ be a $n\times n$ real matrix such that $$ \operatorname{sign}(\sigma) \cdot a_{1,\sigma(1)} a_{2,\sigma(2)} \ldots a_{n,\sigma(n)}\ge 0 $$ for all $\sigma\in S_n$. Is there a name ...
4
votes
0answers
79 views

Methods of solving roots involving matrix determinant

I have a square matrix $F$ whose elements depend non-linearly on a complex parameter $s$. I would like to know the values of $s$ such that $\det(F)=0$, i.e., those $s$ that make $F$ singular. I would ...
4
votes
0answers
107 views

Characterisation of matrices whose real eigenvalues are positive

My question is the following Is there a characterisation of $n\times n$ matrices with real entries whose real eigenvalues are positive? I am interested in this question because I am analysing some ...
4
votes
1answer
123 views

Finding the minimal value of a $4\times 4$ determinant

The question. Let $\xi=(\xi_1,\xi_2,\xi_3,\xi_4)\in\mathbb R^4$ be a vector with irrational coordinates. I am interested in finding the minimal value $\mu_\xi$ of $$\left\vert \det \begin{pmatrix} ...
4
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0answers
93 views

Prove that the matrix $[\Gamma(\lambda_{i}+\mu_{j})]$ is nonsingular.

Let $A$ be an $n\times n$ matrix whose entries are \begin{align*} a_{ij} = [\Gamma(\lambda_{i}+\mu_{j})] \end{align*} where $0 < \lambda_{1} < \ldots < \lambda_{n}$ and $0 < \mu_{1} < \...
4
votes
0answers
67 views

The polarization of the determinant is invariant?

Given $n \in \mathbb N$, I am asked to show that there is a multilinear symmetric $\operatorname{GL}_n$-invariant form $\phi : (M_{n \times n})^l \to \mathbb R$ (for some $l \geq 0$) such that $\phi(A,...
4
votes
0answers
39 views

Sufficient/necessary condition for submatrix determinant (minor) that decreases with size?

Definition. Given a square matrix ${\bf{A}}=[a_{ij}] \in {\mathbb{C}^{n \times n}}$, the submatrix ${{\bf{A}}_{{i_1},{i_2},...{i_k}}}$ is formed by retaining the $({i_1},{i_2},...{i_k})$-th rows and ...
4
votes
0answers
70 views

Find the values of $a$ $\in$ $\mathbb{R}$ where the system $Ax=x$ allows a solution different to the null one

I have to find the values of $a$ $\in$ $\mathbb{R}$ where the system $Ax=x$ allows a solution different to the null one, and then solve the system with those values I found of the following matrix: $...
4
votes
0answers
276 views

Calculation of the Pfaffian of a matrix

I have a set of $N$ numbers $\lbrace \lambda_i\rbrace_{i\in[1,N]}$ that belong to $[0,2\pi[$ and a real number $L$ and I am trying to evaluate the following Pfaffian expression. $$\mathrm{Pf}\left(\...
4
votes
0answers
112 views

How many subsets of $n$ linearly independent binary strings of length $n$?

Let's consider binary words of length $n$ with elements {-1,1}. There are $2^n$ binary words of length $n$. Now let's consider a subset of $n$ such binary words. All possible subsets are $\binom{2^n}{...
4
votes
0answers
318 views

Quadrics intersecting the twisted cubic and a line.

I am trying to understand the determinantal approach on Harris book "Algebraic Geometry: A first course" on proving that the intersection of two quadrics containing the twisted cubic in $\mathbb{P}^3$ ...