Articles of eigenvalues eigenvectors

Prove that a non-positive definite matrix has real and positive eigenvalues

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 […]

Showing that $M$ and $N$ will have same eigenvalues.

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

Eigenvalue problem $y'' + \lambda y = 0,$ $y'(0) = 0$, $y(1) = 0$

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. […]

How to find a eigenvector with a repeated eigenvalue?

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 […]

Is there a connection between the diagonalization of a matrix $A$ and that of the product $DA$ with a diagonal matrix $D$?

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 $DA = P\Lambda P^{-1}$?

Invariant subspaces using matrix of linear operator

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 […]

smallest eigenvalue of rank one matrix minus diagonal

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 […]

Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$

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 […]

do eigenvectors correspond to direction of maximum scaling?

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 : 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 = […]

Priority vector and eigenvectors – AHP method

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 […]