# Thread: trouble with 4x4 matrices

1. ## trouble with 4x4 matrices

find all the eigenvalues and eigenvectors for matrix B

b=
2 0 1 1
0 2 1 1
1 1 2 0
1 1 0 2

i understand how to do 3x3 matrices but cant get my head around 4x4

2. i could solve the polynomial but that would take forever. is there a easier way.

the polynomical i came up with is

$\displaystyle A^4 - 8A^3 + 24A^2 - 32A + 16 = 0$

3. With all of those "2"s along the diagonal, $\displaystyle \lambda$ always appear in "$\displaystyle 2-\lambda$". When I expand it, rather tham multiplying out powers of $\displaystyle 2- \lambda$, I left them as powers and got $\displaystyle (2-\lambda)^4- (2-\lambda)^2= 0$. Letting $\displaystyle x= 2- \lambda$, that equation becomes $\displaystyle x^4- x^2= x^2(x^2- 1)= 0$ which has solutions x= 0 (a double root), x= 1, and x= -1. Since $\displaystyle x= 2-\lambda$, $\displaystyle \lambda= 2$ is a double eigenvalue, and $\displaystyle \lambda= 1$ and $\displaystyle \lambda= 3$ are also eigenvalues.