Lower triangular matrix eigenvectors problem

[eherrtelle59]

Ok, this is starting to come back to me, but I'm stuck again

Homework Statement

M=\begin{bmatrix}
(1-\frac{4}{3}) & 0 \\
-\frac{c}{3} & -c \\
\end{bmatrix}


Find eigenvectors and eigenvalues.

Homework Equations

The Attempt at a Solution

Eigenvalues are [itex] λ_1= (1-\frac{4}{3})>0[/itex] and [itex] λ_2=-c>0 [/itex]

Eigenvector for [itex] λ_2 [/itex] is <0 1>
For [itex]λ_1[/itex], I should get [itex]<(\frac{c}{3}-1) (\frac{4}{3})> [/itex]

However, I end up with (without writing out the matrix again, just giving the equation mind you)

[itex] -c*e_2 = e_1 - \frac{4}{3}*e_1 +\frac{c}{3}*e_1 [/itex]

This is [itex] -c*e_2 = (1-c)*e_1 [/itex]which gets an eigenvector of something like <(1-c) c>

Anyone see my error? Thanks

 

[Office_Shredder]

The clearly negative eigenvalue being allegedly positive is the first thing that struck me as odd, but not really a big deal in the grand scheme of things

I don't understand what you did to get that equation (the left hand side looks like Me₂, and the right hand side is completely unclear to me where it came from, but the fact that you have a scaling of e₁= a scaling of e₂ seems pretty suspicious), or what you are claiming the eigenvector for the first eigenvalue is since it only has one entry as written - is it supposed to be ( (c/3-1), 4/3)?

 

[eherrtelle59]

"The clearly negative eigenvalue being allegedly positive is the first thing that struck me as odd, but not really a big deal in the grand scheme of things"

That's a typo on my part. λ_2 should be negative i.e. -c<0

"is it supposed to be ( (c/3-1), 4/3)?"

Yes

To show explicitly what I did, using the standard eigenvector equation with v_i instead of e_i


\begin{bmatrix}
(1-(4/3)c) & 0 \\
-c/3 & -c \\
\end{bmatrix} *\begin{bmatrix}
v_1 \\
v_2 \\
\end{bmatrix}

= \begin{bmatrix}
(1-4c/3)*v_1 \\
(1-4c/3)*v_1\\
\end{bmatrix}

When you multiply this matrices (as sort of shown...latex skills withstanding) then you get the equation I've written previously.

 

[eherrtelle59]

"but the fact that you have a scaling of e1= a scaling of e2 seems pretty suspicious"

When finding eigenvectors I always get equations like this.

 

[eherrtelle59]

I'm sorry everyone, that should be multiply by

\begin{pmatrix}
(1-4c/3)v_1\\
(1-4c/3)v_2\\
\end{pmatrix}
which does indeed give the right answer,

(c/3)*v_1 = (c/3 -1)*v_2

Sorry for wasting everyone's time, I've finally got it now. Thank you!