Articles of linear algebra

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 $L(x)$? […]

Linear Transformation defined by a Matrix and Invariant Subspaces

I got stuck solving this problem: Let $T:\mathbb{R}^3\to \mathbb{R}^3$ be the linear transformation defined by the matrix A in the standard basis of $\mathbb{R}^3$, $E=\{e_1,e_2,e_3\}$ $$A=\begin{bmatrix} 3 & 1 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{bmatrix}$$ and let $W=\ker(T-3I)$ be a subspace of $\mathbb{R}^3$, Show that there […]

Symbol for Euclidean norm (Euclidean distance)

Which symbol is more commonly used to denote the Euclidean norm: $ \left \| \textbf a \right \| $ or $ \left | \textbf b \right |$?

solutions of a linear equation system

The following matrix is given over $\mathbb R$ A=$\begin{pmatrix} 1 & -1 & -1 \\ 1 & 0 & a \\ 1 & a & 0 \end{pmatrix}$ The linear equation System $Ax=\begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}$ has exactly one solution for all $a$ except $a=-2$ which has no solution. Is it right?

Let $A$ be an 8 x 5 matrix of rank 3, and let $b$ be a nonzero vector in $N(A^T)$. Show $Ax=b$ must be inconsistent.

Here’s the entire question: Let $A$ be an 8 x 5 matrix of rank 3, and let $b$ be a nonzero vector in $N(A^T)$. a) Show that the system $Ax = b$ must be inconsistent. Gonna take a wild stab at this one… If the rank is 3, that means the dimension of the column […]

The variance of the expected distortion of a linear transformation

Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a linear transformation. I am interested in the “average distortion” caused by the action of $A$ on vectors. (i.e stretching or contraction of the norm). Consider the uniform distribution on $\mathbb{S}^{n-1}$, and the random variable $X:\mathbb{S}^{n-1} \to \mathbb{R}$ defined by $X(x)=\|A(x)\|_2^2$. It is easy to see that $E(X)=\frac{1}{n}\sum_{i=1}^n \sigma_i^2$, […]

Good Will Hunting Problem Reasoning

I am trying to understand this problem from the movie Good Will Hunting just because it looks fun. It asks to find the generating function for walks from points i to j. I attached a link here http://www.math.harvard.edu/archive/21b_fall_03/goodwill/ that apparently explains it. I am curious how the geometric series holds true for matrices. I am […]

Sum of two kronecker products as a kronecker product

I seek for the following relationship (if there is one so): $$C \otimes D = (A_1 \otimes B_1) + (A_2 \otimes B_2)$$ I would like to obtain $C = f(A_1,A_2)$ (in terms of $A$’s) and $D = g(B_1,B_2)$ (in terms of $B$’s). For simplicity, we can assume $A_i$ and $B_i$ are covariance matrices, so positive-definite, […]