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Questions tagged [determinant]

Questions about determinants: their computation or their theory. If $E$ is a vector space of dimension $d,ドル then we can compute the determinant of a $d$-uple $(v_1,\ldots,v_d)$ with respect to a basis.

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2 votes
1 answer
80 views

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ドル$ determinant. $$ \begin{bmatrix} A & x & \textbf{0} \\...
0 votes
0 answers
77 views

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 ...
1 vote
0 answers
42 views

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 ...
0 votes
1 answer
63 views

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 ...
0 votes
0 answers
18 views

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 $\det(A+cB),ドル$c \in R,ドル $(b)_{ij}=1 \forall i,j$ ...
3 votes
1 answer
232 views

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

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, ...
4 votes
1 answer
163 views

Let $A,ドル $B,ドル 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 ...
3 votes
1 answer
119 views

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. ...
7 votes
0 answers
176 views

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 $n=7,ドル the 8ドル \times 8$ matrix is $$ M_7 = \begin{pmatrix} 7u & -2 & 0 & ...
1 vote
0 answers
34 views

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 2ドルd\times 2d$ determinant $$ F_d(x;t,u):=\det\begin{pmatrix} ...
13 votes
1 answer
2k views

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_{\...
2 votes
0 answers
37 views

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

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 ...
1 vote
0 answers
38 views

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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