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.

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 […]

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 […]

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 […]

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) $T_{\lambda i}$ […]

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!

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 […]

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 […]

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 […]

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}?$

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