Articles of matrices

How to show $X=\{A\in\mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{Ker}(A)=\{0\}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$?

How to show $X=\{A\in\mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{Ker}(A)=\{0\}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$? Here $\mathcal{L}(\mathbb R^m, \mathbb R^n)$ is the set of all linear applications between $\mathbb R^m$ and $\mathbb R^n$.

What is the enclosed volume of an irregular cube given the x,y,z coordinates of the 8 corners?

I have the xyz coordinates of 8 points that forms an irregular-shaped cube. This is an animation, so the cube is undergoing periodic or cyclical shape-change. The co-planarism of any group or set of any combination of 4 of these points can’t be assumed (it’s a virtual certainty it practically never happens). This question is […]

Fundamental problem of Linear Algebra

What is the fundamental problem of linear algebra? I understand it is a big question and not easy to explain completely, and seems no way to prove an answer is correct. I just wanna listen to you experts’ opinion. For example, can I say that to solve linear equation systems is fundamental in linear algebra? […]

Determinant of an almost-diagonal matrix

I would like to compute the determinant of the $(k+1)\times (k+1)$ matrix below $$J=\begin{vmatrix} y_{k+1}& 0 & \ldots & 0 & y_1 \\ 0& y_{k+1}& \ldots& 0& y_2 \\ \vdots& \vdots& & \vdots &\vdots \\0 & 0&\ldots& y_{k+1} &y_k \\ -y_{k+1} & -y_{k+1} &\ldots &-y_{k+1}& \left(1-y_1-\ldots-y_k \right) \end{vmatrix} $$ The matrix is diagonal, if you […]

Fractional power of matrix

If I am given a matrix, for example $A = \begin{bmatrix} 0.7 & 0.2 & 0.1 \\[0.3em] 0.2 & 0.5 & 0.3 \\[0.3em] 0 & 0 & 1 \end{bmatrix}$, how do I calculate the fractional matrix like $A^{\frac{1}{2}}, A^{\frac{3}{2}}$?

A property of $3 \times 3$ rotation matrices.

Let $A$ be a $3\times 3$ orthogonal matrix with $\det A =1 $, whose angle of rotation is different from $0$ or $\pi$, and let $ M = A -A^t$ -Show that $M$ has rank 2, and that a nonzero vector $X$ in the nullspace of $M$ is an eigenvector of $A$ with eigenvalue 1. […]

Why the pivots and relative to the eigenvalue in symmetric matrix?

In the book, it said, there a quick fast way to test whether the eigenvalue are all positive or not. Just check the pivot of the symmetric matrix, if x no. of positive pivot, it would have x no.of eigenvalue. It also mentioned that it should be prove by the $$A=LDL^T$$ But I can’t understand […]

Differentation Operator

having trouble completing the proof for this question Let $D:\mathbb{R}[X] \to \mathbb{R}[X]$ be the differentiation operator $D(f(X))=f'(X) .$ Prove that $e^{tD}(f(X)) = f(X+t)$ for $t \in \mathbb{R}$ Im having trouble making sense of the question. At first i tried Taylor’s theorem to try and make sense of, and equate the two sides of the equation. […]

Derivative of the determinant of the right stretch tensor

I have to evaluate the derivative $$ \frac{\partial\det\mathcal{U}}{\partial F} $$ where $\mathcal{U}=\sqrt{F^TF}$ and $F$ is a $m\times n$ real matrix. Any suggestion would be appreciated. Thank you all, guys!! You helped me a lot.

trace of product of positive definite matrix

For positive definite matrices $A$ and $C$, positive semidefinite matrices $B$ and $D$, I want to know whether $tr\{ABCD\}=0$ implies that $tr\{BD\}=0$.