Articles of linear algebra

Linear algebra minimal sum

Among all the unit vectors $u=(x,y,z)$ in $\mathbb{R}^3$, find the one for which the sum $x + 2y + 3z$ is minimal. How do I get the minimal? I know the unit vectors of $\mathbb{R}^3$ but do I evaluate one by one to know the minimal or there’s another way? Please help. By the way […]

Eigenvectors of the $2\times2$ zero matrix

I have been given a problem that involves the following matrix: $$\begin{bmatrix}-2 & 0\\0 & -2\end{bmatrix}$$ I calculated the eigenvalues to be $\lambda_{1,2} = -2$ When I go to calculate the eigenvectors I get the following system: $$\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}0 \\ 0\end{bmatrix}$$ The eigenvectors are clearly $\begin{bmatrix}1 \\ 0\end{bmatrix}$ […]

Diagonalization of linear operators

First of all, I´m sorry for my English, I´m Spanish so I hope you can all understand me. Here is my problem. Let $T(p(x))=p(x+1)$ be a linear operator from the space of polynomials with real coefficients and degree less than or equal to $n$. I´m having trouble with the matrix of the operator, and without […]

Matrix sequence convergence vs. matrix power series convergence:

Is my thinking correct? The sequence $A^n$ converges if each entry converges to a finite number. But for a matrix power series, $ I + A + \cdots + A^n + \cdots $ can never converge if it has, for example a “1” in the upper left corner, in entry $a_{11}$. Take, for simplicty, $A$ […]

Does there exist a matrix $P$ such that $P^n=M$ for a special matrix $M$?

Consider the matrix $$ M=\left(\begin{matrix} 0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{matrix}\right). $$ Is there a matrix $P\in{\Bbb C}^{4\times 4}$ such that $P^n=M$ for some $n>1$? One obvious fact is that if such $P$ exists, then $P$ must be nilpotent. However, I have no idea how to deal with this problem. Furthermore, what if $M$ is an […]

Eigenvalues of Augmented Matrix

Let $A$ be an invertible real $n\times n$ matrix and suppose the singular values of $A$ are in the interval $[1,\kappa_2(A)]$, where $\kappa_2(A)$ denotes the (2-norm) condition number of $A$. Consider $$M=\left[\begin{array}{cc}I & A\\ A^T & 0\end{array}\right].$$ Results from various papers I’ve found claim that the eigenvalues of such augmented systems in general (where $I$ […]

Induction for Vandermonde Matrix

Given real numbers $x_1<x_2<\cdots<x_n$, define the Vandermonde matrix by $V=(V_{ij}) = (x^j_i)$. That is, $$V = \left(\begin{array}{cccccc} 1 & x_1 & x^2_1 & \cdots & x^{n-1}_1 & x^n_1 \\ 1 & x_2 & x^2_2 & \cdots & x^{n-1}_2 & x^n_2 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ 1 & x_{n-1}& […]

Where did this matrix come from?

In my lecture my professor spoke of this function $R$ that takes a vector $\vec u=\left(\begin{matrix}a\\b\end{matrix}\right)$ and rotates by $\frac{\pi}{6}$ radians counter clockwise. Then he talked about a matrix $M=\left(\begin{matrix}\frac{\sqrt{3}}{2} & -\frac12\\\frac12 & \frac{\sqrt{3}}{2}\end{matrix}\right)$. He showed finding the lengths on a triangle with a hypotenuse length $1$, and I have a feeling that the $x$ […]

need help Jordan base

Need help, how to find Jordan base for matrix: $A=\begin{pmatrix} -1&-1 &-2 &4 \\ 1&-3 &1 &-2 \\ 0&0&2&-8\\ 0&0 & 2&-6 \end{pmatrix}$ I found the Minimal polynomial: $(x+2)^3$ and normal Jordan is $Aj=\begin{pmatrix} -2&0 &0 &0 \\ 0&-2 &1 &0 \\ 0&0&-2&1\\ 0&0 & 0&-2 \end{pmatrix}$ Then I have problem with find $v_1, v_2, […]

Proof of the Sherman-Morrison Formula

I was reading a few proofs for the Sherman-Morrison Formula, which states that if $A$ is invertible and $M = A + \mathbf{u}\mathbf{v}^T$, then $M^{-1}$ is given by: $$A^{-1} – A^{-1}\mathbf{u} \mathbf{v}^T A^{-1}/(1+\mathbf{v}^TA^{-1}\mathbf{u}).$$ There is a proof (verification) of this on Wikipedia as well as here but both of them do not justify why $(1+\mathbf{v}^TA^{-1}\mathbf{u})$ […]