Questions tagged [determinant]

Question 1

I am reading Multiple View Geometry in Computer Vision and in the chapter 17.1, it talks about the following matrix which needs to have $0$ determinant. $$ \begin{bmatrix} A & x & \textbf{0} \\...

Question 2

This is used in one of the many proofs for the Cayley-Hamilton theorem. My professor noted that this should be proved. However, the proof of this fact is rather straightforward, no? Is the proof I ...

Question 3

Let $R := \mathbb C[x_1,\dots,x_d]/J$ be an affine domain which is the coordinate ring of the affine variety $X = V(J) \subseteq \mathbb C^d$. Let $M \in R^{m\times(n+1)}$ be a matrix with entries ...

Question 4

It is known that, $$ \nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B}) $$ The straightforward way to prove this ...

Question 5

Prove that the determinant of a skew-symmetric matrix of even order does not change if to all its elements we add the same number. i tried calculating $\det(A+cB)$,$c \in R$, $(b)_{ij}=1 \forall i,j$ ...

Question 6

This is a problem from Linear Algebra by Werner H. Greub in Chapter 4: Let $V^*$, $V$ be a pair of dual spaces (${\rm dim}V={\rm dim}V^*=n$) and $\Delta \neq 0$ be a determinant function in $V$. ...

Question 7

In the book that I’m actually using for tensor algebra (second order tensors in $\mathbb{R}^3$), the author defines the cofactor of a tensor as the tensor that transforms the area vector, that is, ...

Question 8

Let $A$, $B$, and $C$ be three $m \times n$ matrices, and define $A(t) = A + t B + t^2C$ as well as $f(t) = \operatorname{rank}(A(t))$. Then $f(t)$ is a constant over $\mathbb{R}$ except finite values ...

Question 9

I am familiar with basic introductory Linear Algebra. I know that the determinant of a 3x3 matrix gives you the volume of the parallelepiped formed by the coordinate vectors after the transformation. ...

Question 10

I want to compute in a nice way the determinant of a $(n+1) \times (n+1)$ tridiagonal matrix $M_n$ for odd $n$. For $n=7$, the $8 \times 8$ matrix is $$ M_7 = \begin{pmatrix} 7u & -2 & 0 & ...

Question 11

Fix an integer $d\ge 2$. Let $x_1,\dots,x_{2d}$ be variables, and let $t_1,\dots,t_{2d}$ and $u_1,\dots,u_{2d}$ be parameters. Consider the $2d\times 2d$ determinant $$ F_d(x;t,u):=\det\begin{pmatrix} ...

Question 12

The determinant of a matrix is given by the Leibniz formula $$\det(A) = \sum_{\tau \in S_n} \text{sgn}(\tau) \prod_{i = 1}^n a_{i\tau(i)} = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{i = 1}^n a_{\...

Question 13

Let $A \in \mathbb{R}^{m, n}$, with $n > m$ and $\text{rank}(A) = m$, and let $\mathcal{B}$ be the set of square $m\times m$ submatrices of $A$. In other words, $\mathcal{B}$ contains all matrices $...

Question 14

In general determinants are not additive (related question). Nevertheless, matrices $A$ and $B$ can be chosen so that $\det A+\det B=\det\left(A+B\right)$ (related answer). For which positive integer ...

Question 15

Let $a_i,b_i,c_i$ be parameters and $x_1,\dots,x_6$ variables. Consider $$ D=\begin{vmatrix} a_1^2(a_1x_1+c_1) & a_1(a_1x_1+c_1) & a_1x_1+c_1 & x_1+b_1 & a_1(c_1-a_1b_1) & c_1-...

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Questions tagged [determinant]

Why is fact that the determinant of this matrix is 0 equivalent to $x'^T Fx=0$

Check proof that the determinant of a polynomial matrix commute with evaluation

Ideal of maximal minors is radical when rank never drops by more than 1 [Reference request]

Is using determinants like this for vector algebra standard?

A quest about the determinant of a skew symmetric matrix [duplicate]

Mathematical Proof from "Linear Algebra" by Werner Greub

Cofactor of a tensor

Rank of $A+tB+t^2C$ is a constant almost everywhere

Area of Parallelogram when Determinant = 0.

Computing the determinant of a tridiagonal matrix

Degree-$k$ part of a structured determinant as a linear combination of minors?

Is there such a mathematical notion as 'antideterminant'?

Non-trivial lower bound for minimum non-zero determinant among square submatrices

Determinants of two matrices and their sum, entries are consecutive primes

Cubic part of a $6\times6$ determinant via $\Delta(ijkm)?$

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