Finding Eigenvalues for a Matrix A

[Lord_Sidious]

Homework Statement

Find the eigenvalues of A, when A =[3,1,1; 0,5,0; -2,0,1]

Homework Equations

The Attempt at a Solution

I took the det(A-$\lambda$I)=0 and found the characteristic polynomial to be:

-$\lambda$$^{3}$+15$\lambda$$^{2}$-73$\lambda$+115

I couldn't figure out whole roots to the equation and wolframalpha gave me
$\lambda$=5, 5-√2, 5+√2

Does this seem like the right answer? I have checked and double checked the determinant. Would I still be able to find a basis for the eigenspace?

 

[DryRun]

For matrix:
$$A= \begin{bmatrix}3 & 1 & 1 \\ 0 & 5 & 0 \\ -2 & 0 & 1\end{bmatrix}$$
Here is the characteristic equation that i get:
$-λ^3+9λ^2-25λ+25=0$

 

[voko]

Because the second column is [0, 5, 0], the polynomial should look like (5 - x)Q(x), where Q(x) is a second-degree polynomial. So it should be very easy to get its roots. Do it.

 

[Lord_Sidious]

For matrix:
$$A= \begin{bmatrix}3 & 1 & 1 \\ 0 & 5 & 0 \\ -2 & 0 & 1\end{bmatrix}$$
Here is the characteristic equation that i get:
$-λ^3+9λ^2-25λ+25=0$

Sorry the matrix was $$A= \begin{bmatrix}3 & 1 & 1 \\ 0 & 5 & 0 \\ -2 & 0 & 7\end{bmatrix}$$
not 2,0,1 in the last row. That was my mistake.

 

[MednataMiza]

Yes, those are the eigenvalues.

 

[Mark44]

Because the second column is [0, 5, 0], the polynomial should look like (5 - x)Q(x), where Q(x) is a second-degree polynomial. So it should be very easy to get its roots. Do it.

The second row is [0, 5, 0].

 

[voko]

The second row is [0, 5, 0].

This is not clear from the original notation. Anyway, that does not matter for determinants so my argument holds.

 

[vela]

Expand the determinant along the second row.

 

[DryRun]

This is not clear from the original notation.

Well, the original notation used is actually very common and can be recognized as a standard, since it is also used in popular software like MATLAB.

 

[voko]

Well, the original notation used is actually very common and can be recognized as a standard, since it is also used in popular software like MATLAB.

Let's put it this way: not clear for me, I always seem to forget the conventions for rows vs columns in a linear transcription. This may stem from the fact that even when working with paper and pencil, I am about equally likely to write down a matrix column after column and row after row. Let's hope this thread will leave a dent in my state of confusion :)

 

[DryRun]

Sorry the matrix was $$A= \begin{bmatrix}3 & 1 & 1 \\ 0 & 5 & 0 \\ -2 & 0 & 7\end{bmatrix}$$
not 2,0,1 in the last row. That was my mistake.

Your characteristic equation from post #1 is correct. Now, you just need to solve it. Usually, you would start by trial and error to find one factor of the polynomial (you already know the easy one is (λ-5)) and then do long division to obtain a quadratic in the quotient, which you would then solve using the quadratic formula.

 

[voko]

find one factor of the polynomial (you already know the easy one is (λ-5)) and then do long division to obtain a quadratic in the quotient

The point I made in #3 is that none of this is required. By tackling the determinant along [0, 5, 0], one gets the polynomial directly in the factored form.