Eigenvalues?

Question

I have a 3 x 3 matrix (B)


2 -8 -11
-2 7 10
2 -6 -9

Solve the characteristic equation for B, and find the eigenvalues.

So far as I can tell the characteristic equation is
[B - lambda (I)]v = 0

I think the diagonal of the matrix B at that point should be:


2 - lambda
7 - lambda
-9 - lambda


I tried the determinant part next, but I just can't get it to work.

Answer 1

xI - B =

x-2 8 11
2 x-7 -10
-2 6 x+9

det(xI - B) = (x -2)[(x-7)(x + 9) + 60] - 8[2(x + 9) - 20] + 11[12 +
+ 2(x - 7)] = (x - 2)(x^2 + 2x - 63 + 60) - 8(2x + 18 - 20) +
+ 11[12 + 2x - 14] = (x - 2)(x^2 + 2x - 3) -8(2x - 2) + 11(2x - 2) = (x - 2)(x - 1)(x + 3) - 16(x - 1) + 22(x - 1) =
(x -1)(x - 2)(x + 3) + 6(x - 1) = (x - 1)[(x -2)(x + 3) + 6] =
= (x - 1)(x^2 + x - 6 + 6) = (x - 1)(x^2 + x) = x(x -1)(x + 1)

Let's find eigenvectors for eigenvalues 0,1,-1

For eigenvector 0:
-------------------------
2x - 8y - 11z = 0
-2x +7y + 10z = 0
2x -6y - 9z = 0

Adding the 1st equation to the 2nd, and subtracting it from the 3rd
2x - 8y - 11z = 0
y+z = 0
2y + 2z = 0

y=-z

putting it into the 1st equation:
2x + 8z - 11z = 0

2x - 3z = 0

2x = 3z

x=1.5z

An eigen vector of 0 is (1.5, -1, 1)

For the eigenvalue 1
----------------------------
2x - 8y - 11z = x
-2x +7y + 10z = y
2x -6y - 9z = z

x -8y - 11z = 0
-2x + 6y + 10z = 0
2x - 6y - 10z = 0

add 2nd row to 3rd
x -8y - 11z = 0
-2x + 6y + 10z = 0

adding twice the 1st row to the 2nd
x -8y - 11z = 0
-10y - 12z = 0

-10y = 12z

y = -1.2z

putting this into the 1st equation:
x -8*(-1.2z) - 11z = 0

x + 9.6z - 11z = 0

x - 1.4z = 0

An eigenvector of 1 is
(1.4, -1.2, 1)

For the eigenvalue -1
-----------------------------
2x - 8y - 11z = -x
-2x +7y + 10z = -y
2x -6y - 9z = -z

3x - 8y - 11z = 0
-2x + 8y +10z = 0
2x - 6y - 8z = 0

Add 2nd equation to the 1st
x - z = 0

put it in the 2nd equation
-2z + 8y + 10z = 0
-8z + 8y = 0
z = - y

An eigen vector of -1 is (1, -1, 1)

Answer 2

The characteristic equation is:
Det[(B - lambda [I]*v] = 0
where Det is the determinant of a matrix.
so first you take matrix B and subtract lambda along its right diagonal.
2-L -8 -11
-2 7-L 10
2 -6 -9-L
where L is the lambda constant
to get the determinant:
(2-L)*(7-L)*(-9-L) + (-2)*(-6)*(-11) + (2)*(-8)*(10)
- (-11)*(7-L)*(2) - (10)*(-6)*(2-L) - (-9-L)*(-8)(-2) = 0
Remember the determinant is the sum of all right diagonal products minus the products of the left diagonals.
a b c
d e f
g h i
for the above matrix this comes out:
Det[A] = a*e*i + d*h*c + g*b*f - c*e*g - f*h*a - i*b*d
if you follow the letters in the matrix you will see how it works.
solve for L here to get lamda, or eigenvalue
I won't actually solve this but it is the answer.