Linear algebra trouble

[phantomAI]

I was wondering if anyone could help me out by better explaining the process in doing each step. I would have gone to my professor but I tend to get lost even more. The book that I have doesn't even help.

Example:
[5 1 2
2 3 2
2 2 4]

Find
*eigenvalues
*eigenvectors
*matrix P and its inverse P^-1 such that P^-1 is a diagonal matrix D.
*exponent e^tM

eigenvectors I usually

[5-x 1 2
2 3-x 2
2 2 4-x]

This isn't a homework question, but I want to know how to do all these. Thanks.
Thanks.

[HallsofIvy]

[5-x 1 2
2 3-x 2
2 2 4-x]

Okay, the determinant, expanding along the first row, is:

(5-x)|3-x 2| - 1|2 2| + 2|2 3-x|
|2 4-x| |2 4-x| |2 2|

= (5-x)((3-x)(4-x)- 4)- 2(4-x)+ 4+ 2(4- 2(3-x))
= (5-x)(x2-7x+ 12)+ 2x- 8+ 4 +9 + 4x- 6
= 5x2- 35x+ 60- x3+7x2- 12x+ 6x- 1
= -x3+ 12x2-41x+ 59= 0.

There is no simple way to solve that equation (the cubic formula is considerably more coplicated than the quadratic formula) but numerical methods give eigenvalues approximately 7.77, 1.52, 2.72.

Now find the eigenvectors By solving
[5 1 2][x] = [7.77x]
[2 3 2][y] [7.77y]
[2 2 4][z] [7.77z]

( Of course, there will be an infinite number of solutions to that. Take a simple one.)

[5 1 2][x] = [1.52x]
[2 3 2][y] [1.52y]
[2 2 4][z] [1.52z]

[5 1 2][x] = [2.72x]
[2 3 2][y] [2.72y]
[2 2 4][z] [2.72z]

Finally, P is the matrix have those vectors as columns.

[phantomAI]

Thanks HallsofIvy