# What is Eigenvectors: Definition and 459 Discussions

In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by

λ

{\displaystyle \lambda }
, is the factor by which the eigenvector is scaled.
Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated.

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1. ### I Unravelling Structure of a Symmetric Matrix

Hey guys, I was wondering if anyone had any thoughts on the following symmetric matrix: $$\begin{pmatrix} 0.6 & 0.2 & -0.2 & -0.6 & -1\\ 0.2 & -0.2 & -0.2 & 0.2 & 1\\ -0.2 & -0.2 & 0.2 & 0.2 & -1\\ -0.6 & 0.2 & 0.2 & -0.6 & 1\\ -1 & 1 & -1 & 1 & -1 \end{pmatrix}$$ Notably, when one solves for...
2. ### Does there exist a 2x2 non-singular matrix with only one 1d eigenspace?

Before going through calculations/reasoning, let me summarize what my questions will be - In order to obtain the desired matrix, I impose five constraints on ##a,b,c,d,## and ##\lambda##. - These five constraints are four equations and an inequality. I am not sure how to work with the...
3. ### I Is there always a matrix corresponding to eigenvectors?

I tried to find the answer to this but so far no luck. I have been thinking of the following: I generate two random vectors of the same length and assign one of them as the right eigenvector and the other as the left eigenvector. Can I be sure a matrix exists that has those eigenvectors?
4. ### I Help in understanding Eigenvectors please

Hi; struggling a little with eigenvectors; I can get to the equation at the foot of the example but I can't understand the "formula" leading to the setting of x = 3 at the foot of the example? thanks martyn
5. ### Find Eigenvalues & Eigenvectors for Exercise 3 (2), Explained!

For exercise 3 (2), , The solution for finding the eigenvector is, However, I am very confused how they got from the first matrix on the left to the one below and what allows them to do that. Can someone please explain in simple terms what happened here? Many Thanks!
6. ### Finding eigenvalues and eigenvectors given sub-matrices

For this, The solution is, However, does someone please know what allows them to express the eigenvector for each of the sub-matrix in terms of t? Many thanks!
7. ### I Find the Eigenvalues and eigenvectors of 3x3 matrix

Assume a table A(3x3) with the following: A [ 1 2 1 ]^T = 6 [ 1 2 1 ]^T A [ 1 -1 1 ]^T = 3 [ 1 -1 1 ]^T A [ 2 -1 0]^T = 3 [ 1 -1 1]^T Find the Eigenvalues and eigenvectors: I have in mind to start with the Av=λv or det(A-λI)v=0.... Also, the first 2 equations seems to have the form Av=λv...
8. ### Mathematica Matrices in Mathematica -- How to calculate eigenvalues, eigenvectors, determinants and inverses?

Hi, In my linear algebra homework, there is a bonus assignment where we are supposed to use Mathematica to calculate matrices and their determinants etc. here is the assignment. Unfortunately, I am a complete newbie when it comes to Mathematica, this is the first time I have worked with...

17. ### Question regarding eigenvectors

So I have been studying for my upcoming math exam and a lot of the problems require to find eigenvalues/eigenvectors.Now the question I have is the following; Take a look at this matrix $$\left[ \begin{matrix} 6 & -3 \\\ 3 & -4 \end{matrix} \right]$$ Now the eigenvalues are...
18. ### I Orthogonality of Eigenvectors of Linear Operator and its Adjoint

Suppose we have V, a finite-dimensional complex vector space with a Hermitian inner product. Let T: V to V be an arbitrary linear operator, and T^* be its adjoint. I wish to prove that T is diagonalizable iff for every eigenvector v of T, there is an eigenvector u of T^* such that <u, v> is...
19. ### Problem calculating eigenvalues and eigenvectors

Hello everyone. I am trying to construct a functioning version of randomfields (specifically 2D_karhunen_loeve_identification_example.py) in Matlab. For that, I have to calculate the Karhunen-Loève expansion of 2D data, since this is what it says in the documentation. I also have some sample...
20. ### Eigenvalues and eigenvectors of J3

The J3 matrix of two dimensional SU2 consists of two row vectors (1 0) and (0 -1). When I calculate the eigenvalues of an eigenvector v the usual way with J3v=kv then I find eigenvalues +-1 and eigenvectors (1 0) and (0 1). But how is it possible to say that there are other eigenvectors and...
21. ### MHB Solving Matrix A: Characteristic Equation and Eigenvectors

good evening everyone! Decided to solve the problems from last year's exams. I came across this example. Honestly, I didn't understand it. Who can help a young student? :) Find characteristic equation of the matrix A in the form of the polynomial of degree of 3 (you do not need to find...
22. ### Finding the directions of eigenvectors symmetric eigenvalue problem

In the symmetric eigenvalue problem, Kv=w^2*v where K~=M−1/2KM−1/2, where K and M are the stiffness and mass matrices respectively. The vectors v are the eigenvectors of the matrix K~ which are calculated as in the example below. How do you find the directions of the eigenvectors? The negatives...
23. ### How do I know if my eigenvectors are right?

For ##M = \begin{pmatrix} 2 & 2\\ 2 & -1 \end{pmatrix}## I found the characteristic equation: ##( λ - 3 )( λ + 2) \therefore λ = 3,-2##Going back we multiply $$\begin{pmatrix} 2 - \lambda & 2\\ 2 & -1 - \lambda \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}$$ Which gives \begin{matrix} 2x -...
24. ### The applications of eigenvectors and eigenvalues | That thing you heard in Endgame has other uses

Zach Star gives an explanation of Eigenvalues and Eigenvectors, with some applications
25. ### Normalisation of eigenvectors convention for exponentiating matrices

Hi, I just have a quick question when I was working through a linear algebra homework problem. We are given a matrix A = \begin{pmatrix} 2 & -2 \\ 1 & -1 \end{pmatrix} and are asked to compute e^{A} . In earlier parts of the question, we prove the identities A = V \Lambda V^{-1} and e^{A}...
26. ### Setting Free variables when finding eigenvectors

upon finding the eigenvalues and setting up the equations for eigenvectors, I set up the following equations. So I took b as a free variable to solve the equation int he following way. But I also realized that it would be possible to take a as a free variable, so I tried taking a as a free...
27. ### Modeling the populations of foxes and rabbits given a baseline

From solving the characteristic equations, I got that ##\lambda = 0.5 \pm 1.5i##. Since using either value yields the same answer, let ##\lambda = 0.5 - 1.5i##. Then from solving the system for the eigenvector, I get that the eigenvector is ##{i}\choose{1.5}##. Hence the complex solution is...