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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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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} & \...
Rofl Ukulus's user avatar
34 votes
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1k views

Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using this formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $$V= \begin{bmatrix} ...
trion's user avatar
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11 votes
1 answer
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Determinant of $n\times n$ matrix with parameter

Problem: Let $\delta \in \mathbb{R}^+$ and $n\in \mathbb{N}$. The matrix $A_n = (a_{i,j}) \in \mathbb{R}^{n\times n}$ is defined as $$ a_{i,j} = \prod_{k=0}^{i-2}\left((j-1)\delta +n-k\right) $...
Sil's user avatar
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10 votes
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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 &...
Graz's user avatar
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10 votes
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601 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 ...
user1071136's user avatar
9 votes
1 answer
257 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})...
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9 votes
1 answer
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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 $A$. ...
Saal Hardali's user avatar
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8 votes
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Geometric proof/evidence for the 3x3 matrix determinant's formula?

I'm struggling to find a visually intuitive proof for the formula of a 3x3 matrix. I searched it online for hours and it seems impossible to find a source that attempt at least to explain where does ...
Gabriele Scarlatti's user avatar
8 votes
2 answers
231 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 \\ ...
HGF's user avatar
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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 ...
mrk's user avatar
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What is the determinant of the matrix of $A_{ij} = \gcd(i,j)$ for $i,j$ ranging from $0$ to $n−1$?

The determinant of the matrix whose entries are $\gcd(i,j)$ for $1≤i,j≤n$ equals $\prod_{k=1}^n \varphi(k)$ where $\varphi$ is Euler's totient function: see A001088 in the OEIS, as well as the paper “...
Gro-Tsen's user avatar
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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 ...
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7 votes
1 answer
198 views

Identically vanishing polynomials in infinite commutative rings

Let $R$ be an infinite, characteristic zero, commutative ring. I can furthermore suppose it is reduced and indecomposable (no nontrivial nilpotents and idempotents). My question is whether there is a ...
Marcus Barão Camarão's user avatar
7 votes
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428 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}(...
Sergio Parreiras's user avatar
7 votes
1 answer
261 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)\...
Jacob's user avatar
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7 votes
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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 ...
anderstood's user avatar
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7 votes
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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}}}...
Nick's user avatar
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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 \...
Erik's user avatar
  • 453
6 votes
0 answers
117 views

Inequality involving a symmetric matrix and 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 ...
meler's user avatar
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any method to estimate determinant of a matrix?

I'm writing a kernel on GPU to compute determinant for my image processing app. The matrices are binary (only have value $[0,1]$), sparse, and have size $32\times32$. Is there any method for ...
dk1111's user avatar
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How to get the characteristic polynomial of this matrix?

Consider a $n\times n$ matrix: $$ M_n = \begin{pmatrix} a_1 & 1 & 0 & 0 & 0 & \cdots & 1 \\ 1 & a_2 & 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & a_3 &...
an offer can't refuse's user avatar
6 votes
0 answers
193 views

binomial determinant

Let $n > 0$, then : $$ \det \left( {2n \choose n+i-j} \right)_{i,j=0}^{n-1} = \prod_{i=0}^{n-1} \frac{2n+i \choose n}{n+i \choose n} $$ The LHS appears to be the determinant of a symetric ...
Pavel's user avatar
  • 101
6 votes
1 answer
878 views

Is there a generalization of Pfaffians?

For an skew-symmetric matrix $A$ (meaning $A^T=-A$), the Pfaffian is defined by the equation $(\text{Pf}\,A)^2=\det A$. It is my understanding that this is defined for anti-symmetric matrices because ...
UglyMousanova19's user avatar
6 votes
0 answers
280 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 (...
vulture's user avatar
  • 403
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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,...
Sarvesh Ravichandran Iyer's user avatar
6 votes
0 answers
103 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 $d$-...
Asaf Shachar's user avatar
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6 votes
0 answers
123 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 $1$-...
amd's user avatar
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6 votes
0 answers
189 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 $B$ ...
okie's user avatar
  • 2,845
6 votes
0 answers
155 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 + (...
Luc's user avatar
  • 61
6 votes
1 answer
856 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 ...
Feng Shuei's user avatar
6 votes
1 answer
6k 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 ...
J.E.M.S's user avatar
  • 2,628
5 votes
0 answers
115 views

Circulant determinant factorization over Z

Let $X_n= \begin{bmatrix} x_1&x_2&\cdots&x_n\\ x_n&x_1&\cdots&x_{n-1}\\ \vdots&\vdots&\ddots&\vdots\\ x_2&x_3&\cdots&x_1\\ \end{bmatrix} $ be a ...
Tomm's user avatar
  • 71
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Understanding $\,\det(A+B)$

From this paper on Determinant of sums, where \begin{equation} \det(A+B) = \sum_{r} \sum_{\alpha,\beta} (-1)^{s(\alpha) + s(\beta)} \det(A[\alpha|\beta])\det(B[\alpha|\beta]), \end{equation} the ...
User101's user avatar
  • 504
5 votes
0 answers
258 views

Connection between Determinant and the Quotient Rule

For the function $\frac{f(x)}{g(x)}$, by the Quotient Rule, $\frac{d}{dx} (\frac{f}{g}) = \dfrac{\frac{df}{dx}g-f \frac{dg}{dx}}{g^2}$. We can write the numerator as $\begin{vmatrix} \frac{df}{dx} &...
Etemon's user avatar
  • 6,716
5 votes
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432 views

Proving that the determinant of a real symplectic matrix is $1$ from its eigenvalues

Let $A$ be a $2n \times 2n$ real symplectic matrix, i.e., $A$ satisfies $$A^TJA = J$$ where $J = \begin{pmatrix} 0 & I_n\\ -I_n & 0 \end{pmatrix}$. It is a well-known fact that $\det(A) = 1$. ...
cosmic_philosopher's user avatar
5 votes
0 answers
81 views

Is there a geometric interpretation for $\det(\nabla X)$?

Let $X$ be a vector field on a Riemannain manifold $M$. Consider $\nabla X:TM \to TM$, where $\nabla$ is the Levi-Civita connection of $M$. We know that $\operatorname{tr}(\nabla X)=\operatorname{div} ...
Asaf Shachar's user avatar
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5 votes
0 answers
114 views

let $A,B$ matrices $n \times n$ and $A \ne B , AB=0$ and $A\ne 0 , B\ne0$ prove $\left|A\right|^2+\left|B\right|^2=0$

let $A,B$ matrices $n \times n$ such that $A \ne B $ and $AB=0$ and $A\ne 0 , B\ne0$ prove $\left|A\right|^2+\left|B\right|^2=0$ My attempt: $$AB=0 \implies \det(AB)=0 \implies \det(A)*\det(B)=0 \...
John caca's user avatar
  • 291
5 votes
0 answers
185 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 ...
Dave's user avatar
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5 votes
0 answers
92 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, $T(1)=1$ $...
rcoder's user avatar
  • 4,545
5 votes
1 answer
104 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: $...
Neisy Sofía Vadori's user avatar
5 votes
1 answer
112 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\ ...
Naweed G. Seldon's user avatar
5 votes
0 answers
358 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 ...
JustThinking's user avatar
5 votes
0 answers
92 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 ...
John M. Campbell's user avatar
5 votes
0 answers
2k 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 ...
Kevin Meier's user avatar
  • 1,545
5 votes
0 answers
129 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) &...
Tyler McMillen's user avatar
5 votes
0 answers
115 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 $\...
darij grinberg's user avatar
5 votes
0 answers
65 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 &...
Tien Kha Pham's user avatar
5 votes
0 answers
300 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 $n, i, j$ ...
Sangchul Lee's user avatar
5 votes
1 answer
320 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,...
k1.M's user avatar
  • 5,466
5 votes
1 answer
239 views

Analytical expression for the determinant of block tridiagonal matrix

I have a $3n\times3n$ matrix $M$ that is in the following block tridiagonal form: $$M=\begin{pmatrix} A & B^T & 0\\ B & A & UBU \\ 0 & UB^T U & A\\ \end{pmatrix}$$ where $A,B,...
JonasB's user avatar
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