Articles of linear algebra

Proof that multilinear rank of tensor less of equal than border rank

Let’s consider the tensor $t \in \mathbb{R}^{d_1 \times d_2 \times d_3}$. It is well-know statement, that multilinear rank of tensor less or equal than border rank, which means that if we consider multilinear rank $rk_{mult} = (rk_1, rk_2, rk_3)$ than $rk_i(t) \le brk(t)$. I have the proof of this simple fact, but I can not […]

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 […]

Intersection of direct sums

Let $A_i, A’_i, i=1\cdots k$ and $I=\operatorname{span}\{v_0\}$ be subspaces of $V$ such that $\operatorname{dim }A_i=\operatorname{dim} A’_i, \forall i$ and $\operatorname{dim} \bigcap_{1}^kA_i=\operatorname{dim} \bigcap_{1}^kA’_i $. The following is true or false :$$\operatorname{dim} \bigcap_{1}^k(A_i\oplus I)=\operatorname{dim} \bigcap_{1}^k(A’_i\oplus I) $$ Thanks.

Gaussian Elimination

Does a simple Gaussian elimination works on all matrices? Or is there cases where it doesn’t work? My guess is yes, it works on all kinds of matrices, but somehow I remember my teacher points out that it doesn’t works on all matrices. But I’m not sure, because I have been given alot of methods, […]

Find angle of incomplete rotation matrix

I’d like to find the angle of rotation of the following matrix $A= \begin{bmatrix} -\frac{1}{3} & \ast & \ast \\ \ast & -\frac{1}{3} & \ast \\ \ast & \ast & -\frac{1}{3} \end{bmatrix}$ I know that because it is an orthogonal matrix there exists some $T \in GL(3, \mathbb{C}$ so that $T^{-1}AT = \begin{bmatrix} \pm 1 […]

are any two vector spaces with the same (infinite) dimension isomorphic?

Is it true that any 2 vector spaces with the same (infinite) dimension are isomorphic? I think that it is true, since we can build a mapping from $V$ to $\mathbb{F}^{N}$ where the cardinality of $N$ is the dimension of the vector space – where by $\mathbb{F}^{N}$ I mean the subset of the full cartesian […]

Nonlinear maps with additivity or homogeneity

This question already has an answer here: How to find a nonlinear function $f:\mathbb{R}^2\to\mathbb{R}^2$ that is almost linear in the sense $f(\alpha (a,b))=\alpha f(a,b)$? [duplicate] 4 answers

Why is rotation about the y axis in $\mathbb{R^3}$ different from rotation about the x and y axis.

In my textbook for a counterclockwise rotation about the x axis we have $\begin{pmatrix} 1 & 0 & 0\\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix}$ For rotation about the z axis we have $\begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 […]

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 […]

Finding the kernel, eigenvalues, and eigenvectors of the operator $L(x) := x'' + 3 x' + 4 x$

I want to find the kernel, eigenvalues and eigenvectors of the differential operator: $$L(x)=x”+3x’-4x$$ on the $\Bbb C \space \space \text{vectorspace} \space \space C^{\infty}(\Bbb R)$ as well as the solution the the homogenous differential equation: $$x”+3x’-4x=0$$ First question: I have only seen differential operators in the from of $\frac{d}{dx}$. Is there something special about $L(x)$? […]