Question

Can a matrix have different eigenvectors for the same eigenvalue?

Answer 1

yes

Answer 2

if you rref it and get more than 1 free variable then it gets 2 Evs

Answer 3

Does that have a special meaning?

Answer 4

im assuming a 3x3 for some reason; but you have as many Evs as free variables

Answer 5

I know for when there is eigenvalue of zero then the matrix is not invertible.

Answer 6

and if you cant get linearly independant Evs the matrix aint diagonlizable

Answer 7

But you are suppose to have one free variable because the characteristic equation must be non trivial solution

Answer 8

you dont always get one free variable

Answer 9

sometimes more, sometimes less

Answer 10

my book gives an example:\[\begin{vmatrix}2&4&3\\
-4&-6&-3\\
3&3&1\end{vmatrix}\]

Answer 11

the charachteristic eq gives us: \(\lambda =1,-2\)

Answer 12

IF an nxn matrix has n distinct lambdas; then it is diagonalizable

Answer 13

But this one doesn't

Answer 14

right, so if you get less then n values, there is a possibility of getting bad vectors to match

Answer 15

How would you get a bad vector? You are given A and you found lambda and now you solve for the eigenvector.

Answer 16

yep

Answer 17

this one for example only results in 2 vectors; but the matrix is 3x3, so we require 3 independant vectors which we fall short of in the end

Answer 18

We need 3 independent vector to disgonalize the matrix right?

Answer 19

for any givn 3x3 matrix; yes

Answer 20

WOW I have the same book!!!

Answer 21

david lay i assume :)

Answer 22

YES YES!!!

Answer 23

you aint in saint leo are you?

Answer 24

No!! liberal arts school

Answer 25

Thanks! for the help!

Answer 26

youre welcome :) good luck