Articles of matrices

How do I rotate a given point on $\mathbb{S}^n$ so that I send it to the South Pole

This seems pretty trivial but I’m not sure what to do. My coordinates are Cartesian, and I want to send point the $(x_1,…,x_{n-1},x_n)$ to point $(0,…,0,-1)$ so that all the other points are also rotated properly.

form of symmetric matrix of rank one

The question is: Let $C$ be a symmetric matrix of rank one. Prove that $C$ must have the form $C=aww^T$, where $a$ is a scalar and $w$ is a vector of norm one. I think we can easily prove that if $C$ has the form $C=aww^T$, then $C$ is symmetric and of rank one. Why […]

Blockwise Symmetric Matrix Determinant

This question arises from another one of mine, but separate enough that I feel it deserves its own thread. Wikipedia says that $$det\begin{bmatrix}A&B\\B &A \end{bmatrix} = det(A+B)det(A-B)$$ Regardless of whether or not A and B commute. Using the general formulation $$det\begin{bmatrix}A&B\\C &D \end{bmatrix} = det(A)det(D – CA^{-1}B)$$ We see that this becomes $$det(AD- ACA^{-1}B)$$ Or […]

Derivative of matrix with respected to vector (matrix in se(3))

We know that the matrix $A \in SE(3)$ can be written in exponential formula, i.e. $$e^{M} = \begin{bmatrix} R_{3\times3} & \vec{t} \\\vec{0}^T & 1 \end{bmatrix}$$ $$M = \begin{bmatrix} [\vec{\omega}]_{\times} & V^{-1}\vec{t} \\\vec{0}^T & 0 \end{bmatrix}$$ ,where $\omega = [\omega_x, \omega_y, \omega_z]^T$ can be retrieved from rodrigues’ formula. and $$V = I_3 + \frac{1-cos{\theta}}{\theta^2}[\vec{\omega}]_{\times} + \frac{\theta […]

$A \in Gl(n,K)$ if and only if $A$ is a product of elementary matrices.

I want to show the statement: $A \in GL(n,K)$ if and only if $A$ is a product of elementary matrices. I could show the reverse implication: Suppose $A=T_1…T_m$ where each $T_k$ is an elementary matrix. An elementary matrix can be of the form: 1) $T_{ij}$ the matrix that swaps the rows $i,j$ 2) $T_{\lambda i}$ […]

Determinant of skew- symmetric matrix

I have to prove that determinant of skew- symmetric matrix of odd order is zero and also that its adjoint doesnt exist. I am sorry if the question is duplicate or already exists.I am not getting any start.I study in Class 11 so please give the proof accordingly. Thanks!

The form of the states on an algebra of $n\times n$ matrices with complex entries

I have a question concerning $n \times n$ matrices. Denote by $M_n(\mathbb{C})$ the algebra of all $n \times n$ matrices with complex entries. Let $\phi$ be a state on $M_n(\mathbb{C})$ i.e. a linear functional $\phi \colon M_n(\mathbb{C}) \rightarrow \mathbb{C}$ such that $\phi(Id)=1$ and $\phi(A^*A) \geq 0$, for each $A \in M_n(\mathbb{C})$. Show that $\phi$ must […]

Does there exist a unique “max” approximator for matrices?

In engineering sometimes a limit of a $p$-norm, (or in practice for some conveniently large $p$): $$\underset{x_k \text{ s.t. }x_k>x_j \forall j}{\underbrace{{\max({\bf x})}}} \approx \underset{p\to \infty}{\lim}\|{\bf x}\|_p=\underset{p\to \infty}{\lim}\left(\sqrt[p]{\sum_{\forall i} ({x_i})^p}\right)$$ is used as a continous approximation to the “max”-function, finding an approximation to the largest scalar $x_k$ in $\bf x$. Could we find some natural […]

Proximal operator of spectral norm of a matrix

How can I calculate the proximal operator of spectral norm for any general matrix, $X\in R^{m\times n}$ i.e., $X^* = \arg \min_X ||X||_2 + \frac{1}{2\tau} ||X-Y||_F^2$ I understand that the proximal operator for nuclear norm $||X||_*$ is computed using Singular Value Thresholding(SVT) algorithm which is similar to $l1$-norm on a vector of singular values. Thus […]

Matrix Notation: What does A = mean?

I am reading about homography in images and such. One thing pops up a lot: $\mathbf{P} = [\mathbf{R}|\mathbf{t}]$ What does this mean? Does this mean: If $\mathbf{R} = \begin{bmatrix}a & b\\\ c &d\end{bmatrix}$ and $ \mathbf{t} = \begin{bmatrix}x\\\ y\end{bmatrix}$, I get $ \mathbf{P} = \begin{bmatrix}a &b &x\\\ c& d& y\end{bmatrix}?$