Here is a quick solution to the eigenvalue problem. Should be refined and extended to calculate eigenvectors.
{0~⍨∈⍵×∘.=⍨⍳≢⍵}({r+.×⍉qt⊣(qt r)←⌹[⎕CT]⍵ ;qt;r}⍣{(|⍺)≡|⍵}) 3 3 ⍴ ⍳9
16.11684397 ¯1.11684397 ¯3.625973215E¯16
As a quick note, my ⎕CT was 1E¯13, however it doesn't seem to zero out the ¯3.625973215E¯16. I suspect this is due to it being a negative number. Its not that big of a hurdle, it can be easily worked around with an extra function. Worlframalpha for example rounds it to 0 in their solution.