# 4x4 Matrix Eigenvalues and Eigenvectors

1. Apr 30, 2015

### jake96

1. The problem statement, all variables and given/known data
I have 4 equations.

3x+6y-5z-t=-8
6x-2y+3z+2t=13
4x-3y-z-3t=-1
5x+6y-3z+4t=-6

I have already solved this matrix using gauss elimination and found that x=1, y=2, z=5, t=-2

Now the next part of the question asks to solve the matrix using eigenvalues and eigenvectors.

2. Relevant equations
Eigenvalues and Eigenvectors

3. The attempt at a solution

I have worked through to find the eigenvalues and received this equation

0=L^4 - 4(L^3) - 30(L^2) - 162L + 537, where L is lander

from this I found the Eigen values
2.2522
8.7456
-3.4989-3.8757i
-3.4989+3.8757i

Which I am fairly confident are correct.
However I am unsure about how to continue from here and how the corresponding eigenvectors will solve the matrix.

Thanks Very Much

2. Apr 30, 2015

### HallsofIvy

Staff Emeritus
Given that matrix A has eigenvalues $\lambda_1$, $\lambda_2$, $\lambda_3$, and $\lambda_ 4$, with corresponding eigenvectors $v_1$, $v_2$, $v_3$, and $v_4$, form the matrix P having those eigenvectors as columns and diagonal matrix D having the eigenvalues on its diagonal. Then $A= PDP^{-1}$. The equation Ax= y is the same as $PDP^{-1}x= y$ and then $DP^{-1}x= P^{1}y$, $P^{-1}x= D^{-1}P^{-1}y$, and, finally, $x= PD^{-1}P^{-1}y$. It is relatively easy to find $P^{-1}$ and $D^{-1}$ is just the diagonal matrix with the reciprocals of the diagonal elements of D on its diagonal.

Last edited: May 2, 2015
3. Apr 30, 2015

### SteamKing

Staff Emeritus
Generally, the Greek letter lambda, λ, (not "lander") is used for the unknown eigenvalues, thus the characteristic equation is written:

λ4 - 4λ3 - 30λ2 - 162λ + 537 = 0

They are.

4. Apr 30, 2015

### jake96

thanks, my bad. for some reason everyone I work with calls it lander instead of lambda

5. Apr 30, 2015

### HallsofIvy

Staff Emeritus
If you are anywhere near Boston, Massachusetts, they may be saying "lamb-der"!

6. May 2, 2015

### Ray Vickson

Does your question go on to explain what it means when it says "solve the matrix"? There are many things you can do with eigenvalues/eigenvectors; youcan use them to solve numerous, varied types of "problems", but I cannot figure out what problem the question is now asking you to solve.