Recent questions in Matrices

Matrices
Answered

Brandon White
2022-11-25

I'd like to expand a real, symmetric and positive definite Matrix $M$ into a Taylor series. I'm trying to do

$M(T)=M({T}_{0})+\frac{\mathrm{\partial}M}{\mathrm{\partial}T}(T-{T}_{0})+\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.

Matrices
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Annie French
2022-11-17

Matrices
Answered

Layton Park
2022-11-17

$I-2\hat{x}{\hat{x}}^{\mathrm{\top}}$

where $\hat{x}$ is a normalized vector, i.e., $\Vert \hat{x}\Vert =1$?

This is always −1 somehow but can't find proof of it anywhere.

Matrices
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Laila Murphy
2022-11-16

Matrices
Answered

Adison Rogers
2022-11-16

$A=\left(\begin{array}{ccc}a+b& a& b\\ b& a& 0\\ a& b& b\end{array}\right)$

Matrices
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Elliana Molina
2022-11-13

$0\le X\le \text{Id},\phantom{\rule{1em}{0ex}}0\le A,$

where $\text{Id}\in {\mathbb{C}}^{n\times n}$ denotes the identity matrix. Is it true that

$XAX\le A,$

or can you give a counterexample?

Matrices
Answered

Keshawn Moran
2022-11-13

(1) Assume that $AB=AC$ for rectangular matrix $A\in {\mathbb{R}}^{a\times b}$ with rank r where $0<r<min(a,b)$

(2) Assume that $({I}_{b}-{A}^{\prime}(A{A}^{\prime}{)}^{-}A)C=0$ where $(A{A}^{\prime}{)}^{-}$ is the pseudo-inverse of $A{A}^{\prime}$

Do (1) and (2) together imply that $(I-{A}^{\prime}({A}^{\prime}A{)}^{-}A)B=0$?

Matrices
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assupecoitteem81
2022-11-11

${x}^{T}y+{y}^{T}z-{x}^{T}z\le 1$

Matrices
Answered

kemecryncqe9
2022-11-11

$det\left(\begin{array}{ccc}{x}^{2}& (x+1{)}^{2}& (x+2{)}^{2}\\ \phantom{\rule{mediummathspace}{0ex}}{y}^{2}& (y+1{)}^{2}& (y+2{)}^{2}\\ \phantom{\rule{mediummathspace}{0ex}}{z}^{2}& (z+1{)}^{2}& (z+2{)}^{2}\end{array}\right)$

Matrices
Answered

Kenna Stanton
2022-11-10

What I did was the following: ${A}^{2}=I;A{A}^{-1}=I\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{A}^{2}=A{A}^{-1}$

$(AB{)}^{2}=ABAB\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]}^{2}=I\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}ABAB=I\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}BA={A}^{-1}{B}^{-1}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}BA=(BA{)}^{-1}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}BA=\left[\begin{array}{cc}\pm 1& 0\\ 0& \pm 1\end{array}\right]$

which I'm not sure whether it's correct or not and if it's correct, which of these ±s to select from.

Matrices
Answered

Paula Cameron
2022-11-07

In Elements of Statistical Learning, we differentiate $RSS(\beta )=(y-X\beta {)}^{T}(y-X\beta )$ w.r.t to $\beta $ to get ${X}^{T}(y-X\beta )$

According to some, this is because

$$(y-X\beta {)}^{T}(y-X\beta )={y}^{T}y-2{\beta}^{T}{X}^{T}y+{\beta}^{T}{X}^{T}X\beta $$

Matrices
Answered

drogaid1d8
2022-11-04

Define

$${M}_{k}={\int}_{-\frac{1}{2}}^{\frac{1}{2}}{t}^{k}L(t)dt$$

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

$$\u27e8{M}_{2}x,x\u27e9\ge \u27e8{M}_{1}{M}_{0}^{-1}{M}_{1}x,x\u27e9$$

This is obviously true for n=1 since by Cauchy-Schwarz inequality we have

$$(}{\int}_{-\frac{1}{2}}^{\frac{1}{2}}tL(t)dt{{\textstyle )}}^{2}\le {\textstyle (}{\int}_{-\frac{1}{2}}^{\frac{1}{2}}{t}^{2}L(t)dt{\textstyle )}{\textstyle (}{\int}_{-\frac{1}{2}}^{\frac{1}{2}}L(t)dt{\textstyle )$$

That is ${M}_{1}^{2}\le {M}_{2}{M}_{0}.$

Matrices
Answered

MISA6zh
2022-11-04

$${M}_{ij}={\rho}^{|i-j|}$$

for some $\rho \in (0,1)$. 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.

Matrices
Answered

Deja Bradshaw
2022-10-29

Matrices
Answered

Gerardo Aguilar
2022-10-27

$$\theta {C}^{2}=C$$

How to show that all the diagonal entries of C are equal and so offdiagonal entries of C are also equal?

Matrices
Answered

Ralzereep9h
2022-10-25

Matrices
Answered

sorrowandsongto
2022-10-23

Matrices
Answered

link223mh
2022-10-23

$$1={1}^{n}$$

for all $n\ge 0$, and,

$$AB\oplus CD=(A\oplus C)(B\oplus D)$$

for all A,B,C,D, we have:

$$\begin{array}{rl}{e}^{X}\oplus 1& =\left(\sum _{n=0}^{\mathrm{\infty}}\frac{{X}^{n}}{n!}\right)\oplus 1\\ & =\sum _{n=0}^{\mathrm{\infty}}(\frac{{X}^{n}}{n!}\oplus {1}^{n})\\ & =\sum _{n=0}^{\mathrm{\infty}}\frac{(X\oplus 1{)}^{n}}{n!}\\ & ={e}^{X\oplus 1}.\end{array}$$

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