Matrices questions and answers

Recent questions in Matrices
shatichome 2022-11-25

Answer this equation ${x}^{2}-1=0$

Brandon White 2022-11-25

How do I calculate the derivative of matrix?I'd like to expand a real, symmetric and positive definite Matrix $M$ into a Taylor series. I'm trying to do$M\left(T\right)=M\left({T}_{0}\right)+\frac{\mathrm{\partial }M}{\mathrm{\partial }T}\left(T-{T}_{0}\right)+\cdots$$T$ is a algebraic vector of parameters (e.g. temperatures at finite element nodes). I'm only interested in the first order term, i.e. the derivative of $M$ at ${T}_{0}$. My Professor tried to do this, but used unclear notation.

Annie French 2022-11-17

If we let R be the ring of $2×2$ complex matrices. When is the left annihilator just equal to {0}? I see that if A is invertible ${\text{Ann}}_{R}\left(A\right)$ is trivial since if $M\in {\text{Ann}}_{R}\left(A\right)$ then $MA=0$ so we can just multiply on the right by ${A}^{-1}$ and so $M=0$

Layton Park 2022-11-17

What is the determinant of the following Householder matrix$I-2\stackrel{^}{x}{\stackrel{^}{x}}^{\mathrm{\top }}$where $\stackrel{^}{x}$ is a normalized vector, i.e., $‖\stackrel{^}{x}‖=1$?This is always −1 somehow but can't find proof of it anywhere.

Laila Murphy 2022-11-16

Determine as the parameters $a,b\in \mathbb{R}$ the rank of the following matrix$A=\left(\begin{array}{ccc}a+b& a& b\\ b& a& 0\\ a& b& b\end{array}\right)$

Elliana Molina 2022-11-13

Let $X,A\in {\mathbb{C}}^{n×n}$ and suppose$0\le X\le \text{Id},\phantom{\rule{1em}{0ex}}0\le A,$where $\text{Id}\in {\mathbb{C}}^{n×n}$ denotes the identity matrix. Is it true that$XAX\le A,$or can you give a counterexample?

Keshawn Moran 2022-11-13

Suppose I have matrix A and vectors B and C. I assume two facts about these matrices:(1) Assume that $AB=AC$ for rectangular matrix $A\in {\mathbb{R}}^{a×b}$ with rank r where $0(2) Assume that $\left({I}_{b}-{A}^{\prime }\left(A{A}^{\prime }{\right)}^{-}A\right)C=0$ where $\left(A{A}^{\prime }{\right)}^{-}$ is the pseudo-inverse of $A{A}^{\prime }$Do (1) and (2) together imply that $\left(I-{A}^{\prime }\left({A}^{\prime }A{\right)}^{-}A\right)B=0$?

assupecoitteem81 2022-11-11

Let $x,y,z$ are three $n×1$ vectors. For each vector, every element is between 0 and 1, and the sum of all elements in each vector is 1. Why the following inquality holds:${x}^{T}y+{y}^{T}z-{x}^{T}z\le 1$

kemecryncqe9 2022-11-11

Find the value of the following determinant.$det\left(\begin{array}{ccc}{x}^{2}& \left(x+1{\right)}^{2}& \left(x+2{\right)}^{2}\\ \phantom{\rule{mediummathspace}{0ex}}{y}^{2}& \left(y+1{\right)}^{2}& \left(y+2{\right)}^{2}\\ \phantom{\rule{mediummathspace}{0ex}}{z}^{2}& \left(z+1{\right)}^{2}& \left(z+2{\right)}^{2}\end{array}\right)$

Kenna Stanton 2022-11-10

If ${A}^{2}=I,{B}^{2}=\left[\begin{array}{cc}3& 2\\ -2& -1\end{array}\right],AB=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, find BA.What I did was the following: ${A}^{2}=I;A{A}^{-1}=I\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}{A}^{2}=A{A}^{-1}$$\left(AB{\right)}^{2}=ABAB\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}{\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]}^{2}=I\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}ABAB=I\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}BA={A}^{-1}{B}^{-1}\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}BA=\left(BA{\right)}^{-1}\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}BA=\left[\begin{array}{cc}±1& 0\\ 0& ±1\end{array}\right]$which I'm not sure whether it's correct or not and if it's correct, which of these ±s to select from.

Paula Cameron 2022-11-07

Multiplying Matrix X by ${X}^{T}=2{X}^{T}$?In Elements of Statistical Learning, we differentiate $RSS\left(\beta \right)=\left(y-X\beta {\right)}^{T}\left(y-X\beta \right)$ w.r.t to $\beta$ to get ${X}^{T}\left(y-X\beta \right)$According to some, this is because$\left(y-X\beta {\right)}^{T}\left(y-X\beta \right)={y}^{T}y-2{\beta }^{T}{X}^{T}y+{\beta }^{T}{X}^{T}X\beta$

drogaid1d8 2022-11-04

Let $L:\left(-\frac{1}{2},\frac{1}{2}\right)\to M\left(n×n\right)$ be a continuous matrix-valued function such that each L(t) is symmetric and positive definite.Define${M}_{k}={\int }_{-\frac{1}{2}}^{\frac{1}{2}}{t}^{k}L\left(t\right)dt$I want to prove or disprove that${M}_{2}\ge {M}_{1}{M}_{0}^{-1}{M}_{1}$In other words, for all $x\in {\mathbb{R}}^{n}$ we have$⟨{M}_{2}x,x⟩\ge ⟨{M}_{1}{M}_{0}^{-1}{M}_{1}x,x⟩$This is obviously true for n=1 since by Cauchy-Schwarz inequality we have$\left({\int }_{-\frac{1}{2}}^{\frac{1}{2}}tL\left(t\right)dt{\right)}^{2}\le \left({\int }_{-\frac{1}{2}}^{\frac{1}{2}}{t}^{2}L\left(t\right)dt\right)\left({\int }_{-\frac{1}{2}}^{\frac{1}{2}}L\left(t\right)dt\right)$That is ${M}_{1}^{2}\le {M}_{2}{M}_{0}.$

MISA6zh 2022-11-04

I have a matrix representation $\varphi$: $G\to GL\left(d,\mathbb{C}\right)$ and make a direct sum with its complex conjugate, how do I prove that this sum is matrix representation of G over $\mathbb{R}$? I would perhaps prove equivalence with some real matrix representation $\psi$: $\varphi$: $G\to GL\left(2d,\mathbb{R}\right)$, but just don´t know the steps.

Gerardo Aguilar 2022-10-27

If $\theta >0$ and ${C}_{n×n}$ is a symmetric matrix of rank $n-1$ (with $\stackrel{\to }{\mathbf{1}}$ as the only vector in kernel) such that$\theta {C}^{2}=C$How to show that all the diagonal entries of C are equal and so offdiagonal entries of C are also equal?

Ralzereep9h 2022-10-25

If H is a orthonormal matrix and u is a vector such that $||u|{|}_{2}=1$, does there exist some C such that $||Hu|{|}_{\mathrm{\infty }}\le C||u|{|}_{\mathrm{\infty }}$?

sorrowandsongto 2022-10-23