Finding an eigenvector for a specific eigenvalue of a symbolic matrix

Question

The relevant matrix is $$M=\left( \begin{matrix} p^3+\frac{1}{\sqrt{3}}p^8 & p^1-ip^2 & p^4-ip^5 \\ p^1 +i p^2 & -p^3 + \frac{1}{\sqrt{3}}p^8 & p^6-ip^7\\ p^4+i p^5 & p^6+i p^7 & -\frac{2}{\sqrt{3}}p^8\end{matrix} \right)$$

{{p3 + p8/Sqrt[3], p1 - I p2, 
  p4 - I p5}, {p1 + I p2, -p3 + p8/Sqrt[3], p6 - I p7}, {p4 + I p5, 
  p6 + I p7, -((2 p8)/Sqrt[3])}}

using the Eigenvectors[] function returns only zero and an error "unable to find all eigenvectors". I have separately found the eigenvalues $\mu_i$ by solving the cubic characteristic polynomial. This was surely half of the difficulty of the problem. To find the corresponding eigenvector should simply be a problem of Gaussian elimination on $(M-\mu_1I)\vec{v}=0$

My question is, what is the most streamlined way to now find the eigenvector corresponding to a specific e-val, say $\mu_1$? (I haven't included the explicit expression for $\mu_i$ as they are lengthy and it seems to me the problem can be equally solved while keeping it abstract.)

Answer 1

On Mathematica 12.1 I can get the Eigenvectors and Eigenvalues. I noticed each eigenvalue was 1/3 times some Root expression so I created some replacement rules to express the eigenvectors in terms of the $\mu_i$

m = {{p3 + p8/Sqrt[3], p1 - I p2, 
    p4 - I p5}, {p1 + I p2, -p3 + p8/Sqrt[3], p6 - I p7}, {p4 + I p5, 
    p6 + I p7, -((2 p8)/Sqrt[3])}};
eval = Eigenvalues[m];
rules = {eval[[1, 2]] :> 3 μ1, eval[[2, 2]] :> 3 μ2, 
   eval[[3, 2]] :> 3 μ3};
evec = Eigenvectors[m];
FullSimplify[evec /. rules]

$$ \left( \begin{array}{ccc} \frac{(\text{p1}-i \text{p2}) \left(3 \text{$\mu $1}+2 \sqrt{3} \text{p8}\right)+3 (\text{p4}-i \text{p5}) (\text{p6}+i \text{p7})}{3 (\text{p1}-i \text{p2}) (\text{p4}+i \text{p5})-(\text{p6}+i \text{p7}) \left(-3 \text{$\mu $1}+3 \text{p3}+\sqrt{3} \text{p8}\right)} & \frac{3 \text{$\mu $1}^2-3 \text{$\mu $1} \text{p3}-2 \sqrt{3} \text{p3} \text{p8}-3 \text{p4}^2-3 \text{p5}^2-2 \text{p8}^2+\sqrt{3} \text{$\mu $1} \text{p8}}{3 (\text{p1}-i \text{p2}) (\text{p4}+i \text{p5})-(\text{p6}+i \text{p7}) \left(-3 \text{$\mu $1}+3 \text{p3}+\sqrt{3} \text{p8}\right)} & 1 \\ \frac{(\text{p1}-i \text{p2}) \left(3 \text{$\mu $2}+2 \sqrt{3} \text{p8}\right)+3 (\text{p4}-i \text{p5}) (\text{p6}+i \text{p7})}{3 (\text{p1}-i \text{p2}) (\text{p4}+i \text{p5})-(\text{p6}+i \text{p7}) \left(-3 \text{$\mu $2}+3 \text{p3}+\sqrt{3} \text{p8}\right)} & \frac{3 \text{$\mu $2}^2-3 \text{$\mu $2} \text{p3}-2 \sqrt{3} \text{p3} \text{p8}-3 \text{p4}^2-3 \text{p5}^2-2 \text{p8}^2+\sqrt{3} \text{$\mu $2} \text{p8}}{3 (\text{p1}-i \text{p2}) (\text{p4}+i \text{p5})-(\text{p6}+i \text{p7}) \left(-3 \text{$\mu $2}+3 \text{p3}+\sqrt{3} \text{p8}\right)} & 1 \\ \frac{(\text{p1}-i \text{p2}) \left(3 \text{$\mu $3}+2 \sqrt{3} \text{p8}\right)+3 (\text{p4}-i \text{p5}) (\text{p6}+i \text{p7})}{3 (\text{p1}-i \text{p2}) (\text{p4}+i \text{p5})-(\text{p6}+i \text{p7}) \left(-3 \text{$\mu $3}+3 \text{p3}+\sqrt{3} \text{p8}\right)} & \frac{3 \text{$\mu $3}^2-3 \text{$\mu $3} \text{p3}-2 \sqrt{3} \text{p3} \text{p8}-3 \text{p4}^2-3 \text{p5}^2-2 \text{p8}^2+\sqrt{3} \text{$\mu $3} \text{p8}}{3 (\text{p1}-i \text{p2}) (\text{p4}+i \text{p5})-(\text{p6}+i \text{p7}) \left(-3 \text{$\mu $3}+3 \text{p3}+\sqrt{3} \text{p8}\right)} & 1 \\ \end{array} \right) $$ Copy-able result below:

{{(3 (p4 - I p5) (p6 + I p7) + (p1 - I p2) (2 Sqrt[3] p8 + 
      3 μ1))/(
  3 (p1 - I p2) (p4 + I p5) - (p6 + I p7) (3 p3 + Sqrt[3] p8 - 
      3 μ1)), (-3 p4^2 - 3 p5^2 - 2 Sqrt[3] p3 p8 - 2 p8^2 - 
   3 p3 μ1 + Sqrt[3] p8 μ1 + 3 μ1^2)/(
  3 (p1 - I p2) (p4 + I p5) - (p6 + I p7) (3 p3 + Sqrt[3] p8 - 
      3 μ1)), 1}, {(
  3 (p4 - I p5) (p6 + I p7) + (p1 - I p2) (2 Sqrt[3] p8 + 3 μ2))/(
  3 (p1 - I p2) (p4 + I p5) - (p6 + I p7) (3 p3 + Sqrt[3] p8 - 
      3 μ2)), (-3 p4^2 - 3 p5^2 - 2 Sqrt[3] p3 p8 - 2 p8^2 - 
   3 p3 μ2 + Sqrt[3] p8 μ2 + 3 μ2^2)/(
  3 (p1 - I p2) (p4 + I p5) - (p6 + I p7) (3 p3 + Sqrt[3] p8 - 
      3 μ2)), 1}, {(
  3 (p4 - I p5) (p6 + I p7) + (p1 - I p2) (2 Sqrt[3] p8 + 3 μ3))/(
  3 (p1 - I p2) (p4 + I p5) - (p6 + I p7) (3 p3 + Sqrt[3] p8 - 
      3 μ3)), (-3 p4^2 - 3 p5^2 - 2 Sqrt[3] p3 p8 - 2 p8^2 - 
   3 p3 μ3 + Sqrt[3] p8 μ3 + 3 μ3^2)/(
  3 (p1 - I p2) (p4 + I p5) - (p6 + I p7) (3 p3 + Sqrt[3] p8 - 
      3 μ3)), 1}}

Edit: the eigenvalues look like this:

{1/3 Root[-27 p3 p4^2 - 27 p3 p5^2 - 54 p1 p4 p6 - 54 p2 p5 p6 + 
     27 p3 p6^2 + 54 p2 p4 p7 - 54 p1 p5 p7 + 27 p3 p7^2 - 
     18 Sqrt[3] p1^2 p8 - 18 Sqrt[3] p2^2 p8 - 18 Sqrt[3] p3^2 p8 + 
     9 Sqrt[3] p4^2 p8 + 9 Sqrt[3] p5^2 p8 + 9 Sqrt[3] p6^2 p8 + 
     9 Sqrt[3] p7^2 p8 + 
     6 Sqrt[3]
       p8^3 + (-9 p1^2 - 9 p2^2 - 9 p3^2 - 9 p4^2 - 9 p5^2 - 9 p6^2 - 
        9 p7^2 - 9 p8^2) #1 + #1^3 &, 1], 
 1/3 Root[-27 p3 p4^2 - 27 p3 p5^2 - 54 p1 p4 p6 - 54 p2 p5 p6 + 
     27 p3 p6^2 + 54 p2 p4 p7 - 54 p1 p5 p7 + 27 p3 p7^2 - 
     18 Sqrt[3] p1^2 p8 - 18 Sqrt[3] p2^2 p8 - 18 Sqrt[3] p3^2 p8 + 
     9 Sqrt[3] p4^2 p8 + 9 Sqrt[3] p5^2 p8 + 9 Sqrt[3] p6^2 p8 + 
     9 Sqrt[3] p7^2 p8 + 
     6 Sqrt[3]
       p8^3 + (-9 p1^2 - 9 p2^2 - 9 p3^2 - 9 p4^2 - 9 p5^2 - 9 p6^2 - 
        9 p7^2 - 9 p8^2) #1 + #1^3 &, 2], 
 1/3 Root[-27 p3 p4^2 - 27 p3 p5^2 - 54 p1 p4 p6 - 54 p2 p5 p6 + 
     27 p3 p6^2 + 54 p2 p4 p7 - 54 p1 p5 p7 + 27 p3 p7^2 - 
     18 Sqrt[3] p1^2 p8 - 18 Sqrt[3] p2^2 p8 - 18 Sqrt[3] p3^2 p8 + 
     9 Sqrt[3] p4^2 p8 + 9 Sqrt[3] p5^2 p8 + 9 Sqrt[3] p6^2 p8 + 
     9 Sqrt[3] p7^2 p8 + 
     6 Sqrt[3]
       p8^3 + (-9 p1^2 - 9 p2^2 - 9 p3^2 - 9 p4^2 - 9 p5^2 - 9 p6^2 - 
        9 p7^2 - 9 p8^2) #1 + #1^3 &, 3]}