 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

### Let $A,B$ be square matrices. $A\ne 0$ (as a matrix) and ${A}^{2}=AB$. How can I prove that $|\left(A-B\right)|=0$? I think that key is a chain $|\left(A-B\right)|=\cdots =|\left(B-A\right)|$ but not sure if it's the right way. And prove that 0 is A-eigenvalue. Adison Rogers 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

### Suppose that M is an $n×n$ matrix with entries given by${M}_{ij}={\rho }^{|i-j|}$for some $\rho \in \left(0,1\right)$. Is it true that the matrix M is non-negative definite (or even positive definite)?I was trying to write M as ${A}^{\mathrm{\top }}A$ for some matrix A, but could not do so. Any help would be greatly appreciated. Deja Bradshaw 2022-10-29

### 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

### If A is a complex matrix of order n and A is nilpotent, that is, there exists a positive integer s such that ${A}^{s}=0$. Let's say that ${e}^{A}=\sum _{k=0}^{\mathrm{\infty }}\frac{{A}^{k}}{k!}$. Prove that ${e}^{A}$ is similar to ${I}_{n}+A$, where ${I}_{n}$ is the identity matrix of order n. link223mh 2022-10-23
### Consider a direct sum vector space $V={V}_{1}\oplus {V}_{2}$, and let X be a matrix acting on ${V}_{1}$. I recently derived the equation in the title of this question, where 1 denotes identity matrix on ${V}_{2}$, in the following way. Since$1={1}^{n}$for all $n\ge 0$, and,$AB\oplus CD=\left(A\oplus C\right)\left(B\oplus D\right)$for all A,B,C,D, we have:$\begin{array}{rl}{e}^{X}\oplus 1& =\left(\sum _{n=0}^{\mathrm{\infty }}\frac{{X}^{n}}{n!}\right)\oplus 1\\ & =\sum _{n=0}^{\mathrm{\infty }}\left(\frac{{X}^{n}}{n!}\oplus {1}^{n}\right)\\ & =\sum _{n=0}^{\mathrm{\infty }}\frac{\left(X\oplus 1{\right)}^{n}}{n!}\\ & ={e}^{X\oplus 1}.\end{array}$ benatudq 2022-10-23