Finding the eigenvectors in triangular matrices

[eherrtelle59]

I thought I would ask this in the homework section.

Homework Statement

I should be able to write down the eigenvectors and eigenvalues of diagonal and triangular matrices on sight.


M = \begin{bmatrix}
1 &0 \\[0.3em]
0 & x \\[0.3em]

\end{bmatrix}

vs.

M = \begin{bmatrix}
-1 & -1 \\[0.3em]
0 & x-(1/4) \\[0.3em]

\end{bmatrix}

Homework Equations

The Attempt at a Solution

Obviously, I can see in the diagonal matrix that eigenvalues are λ_1 =1 and λ_2 =x
Therefore, the eigenvectors are e_1= (1 0) and e_2 =(0 1)

Now for the triangular matrix: by sight, I can see λ_1 =-1 and e_1 = (1 0 )

What I cannot do is figure out (by looking at the matrix) that if λ_2=x-(1/4) then e_2 = (-1 x+(3/4) ) Any help would be much appreciated. Thanks!

 

[tiny-tim]

hi eherrtelle59!

(try using the X₂ button just above the Reply box )

What I cannot do is figure out (by looking at the matrix) that if λ₂=x-(1/4) then e₂ = (-1 x+(3/4) )

-1*-1 + -1*(x+3/4) = (x - 1/4)*-1

0*-1 + (x-1/4)*(x+3/4) = (x - 1/4)*(x+3/4)

 

[Mark44]

I thought I would ask this in the homework section.

This looks very similar to a thread you started in one of the math technical sections (https://www.physicsforums.com/showthread.php?t=602860). It's slightly different, so I'm just going to advise you that homework and homework-like questions belong here, not in the math technical section.