I have a $2\times2$ matrix $J$ of rank $2$ and a $2\times2$ diagonal positive definite matrix $Α$. Denote by $J^+$ a pseudoinverse of $J$. I can find many counterexamples for which $J^+AJ$ is not positive definite (e.g. $J=\left(\begin{smallmatrix}2&1\\1&1\end{smallmatrix}\right)$ and $A=\left(\begin{smallmatrix}1&0\\0&2\end{smallmatrix}\right)$), but for all of them $J^+AJ$ has real and positive eigenvalues. So, I was wondering […]

This question already has an answer here: If $ \mathrm{Tr}(M^k) = \mathrm{Tr}(N^k)$ for all $1\leq k \leq n$ then how do we show the $M$ and $N$ have the same eigenvalues? 2 answers

Find the eigenvalues of $$y” + \lambda y = 0, \; y'(0) = 0, y(1) = 0$$ For $\lambda >0$, $$y(x) = c_1 \cos(\sqrt{\lambda} x) + c_2 \sin(\sqrt{\lambda}x)$$ We get that $y'(0) = 0 \implies c_2 = 0$, but when I try to solve for $\lambda$ when doing $y(1) = 0$, I run into trouble. […]

The eigenvalues of my matrix are $x_1= 1$ and $x_2=3$ I get an eigenvector $V = t~[ 4~~~~~~ 3 ~~~~~1 ]^T $ but how can I diagonalize the matrix if I have the same column repeated twice. Should I just use different values for t since they are all eigenvectors of the same matrix? matrix […]

Given a diagonalizable matrix $A = P_0\Lambda_0 P_0^{-1}$ and a diagonal matrix $D$ with $\det D=1$, is there any connection between $P_0$ and the matrix $P$ of the diagonalization of $DA = P\Lambda P^{-1}$?

I am attempting the following problem but stuck at some parts: How does one find the (2 dimensional) subspaces that are invariant under $A$ for $$A = \begin{pmatrix} 1 & 0 & 0 \\ 0 &2 & 0\\ 0 & 0 & 3\\ \end{pmatrix}\ \in M_{3} (\mathbb{R}).$$ Solution: I found the 1-dimensional subspaces: They are […]

Let $x$ be a $d$-dimensional real vector with $\| x\| = 1$. Define $X := xx^T – \mathrm{diag}(xx^T)$. Is it possible to show that $\lambda_{\mathrm{min}}( X ) \geq – 1/2$? Running a bunch of random trials in python seems to suggest this is true, but I’m not sure how to show it. The best I […]

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three different matrices in $\Bbb{R}^2$ and sometimes in $\Bbb{C}^2$ which represent a part of the control system and each matrix has just two […]

Does the eigenvector correspond to a direction in which maximum scaling occurs by a given transformation matrix (A) acting upon this vector. I quote from : https://math.stackexchange.com/q/243553 No other vector when acted by this matrix will get stretched as much as this eigenvector. Is the above statement always true?… For example let $$ A = […]

I’m reading about the AHP method (Analytic Hierarchy Process). On page 2 of this document, it says: Given the priorities of the alternatives and given the matrix of preferences for each alternative over every other alternative, what meaning do we attach to the vector obtained by weighting the preferences by the corresponding priorities of the […]

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