<s>[INST] Eigenvalues of companion matrix of $4x^3 - 3x^2 + 9x - 1$ I want to find all the roots of a polynomial and decided to compute the eigenvalues of its companion matrix.
How do I do that?
For example, if I have this polynomial: $4x^3 - 3x^2 + 9x - 1$, I compute the companion matrix:
$$\begin{bmatrix} 0&0&\frac{3}{4} \\ 1&0&-\frac{9}{4} \\ 0&1&\frac{1}{4} \end{bmatrix}$$
Now how can I find the eigenvalues of this matrix? Thanks in advance.  
 [/INST] Hey There, so if I am assuming correctly for your case, you want to find eigenvalues for this matrix, which is essentially solving for your roots of the characteristic polynomial of the matrix after doing the determinant operation on it. So to go off from Robert idea, we want to use the equation,
det(A$-\lambda$ I) = $0$ $~~~$(following from this we can plug in the coefficient matrix given).
det(A$-\lambda$ I) = 
$\left[\begin{array}{ccc}
0-\lambda & 0 & \dfrac{3}{4} \\ 
1 & 0-\lambda & -\dfrac{9}{4} \\ 
0 & 1 & \dfrac{1}{4}-\lambda
\end{array} \right] = 0$, where A is your coefficient matrix and I is the identity matrix.
I = $\left[\begin{array}{ccr}
1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1
\end{array} \right]$
From here we can now find out the eigenvalues of the matrix A as follows:
$\underline {\text{Evaluation of the determinant expanding it by the minors of column 1:}}$
= $~-\lambda
\left[\begin{array}{cc}
            -\lambda & -\dfrac{9}{4} \\ 
            1 & \dfrac{1}{4}-\lambda  
\end{array} \right]
-1
\left[\begin{array}{cc}
            0 & \dfrac{3}{4} \\ 
            1 & \dfrac{1}{4}-\lambda  
\end{array} \right]
+ 0
\left[\begin{array}{rr}
            0 & \dfrac{3}{4} \\ 
            -\lambda & -\dfrac{9}{4}  
\end{array} \right]
$ 
$\Rightarrow ~ -\lambda
\left[\begin{array}{c}
            \lambda^{2} -\dfrac{1}{4}\lambda + \dfrac{9}{4} \\   
\end{array} \right]
-1
\left[\begin{array}{cc}
            0 -\dfrac{3}{4} \\   
\end{array} \right] 
+ ~0
\left[\begin{array}{rr}
            0 + \dfrac{3}{4}\lambda \\   
\end{array} \right]
$ 
$\Rightarrow ~$ $-\lambda^3+\dfrac{1}{4}\lambda^2-\dfrac{9}{4}\lambda+\dfrac{3}{4}$, $~$ Hence our characteristic polynomial is now obtained. 
$$P(\lambda)=-\lambda^3+\dfrac{1}{4}\lambda^2-\dfrac{9}{4}\lambda+\dfrac{3}{4}$$
If you need assistance on how to find the characteristic polynomial by evaluating the determinant, here is a reference: Computing Determinants
\
After solving this polynomial for its roots (eigenvalues) we get the following: 
{$\lambda = (0.329,0.000) ~~ \lambda = (-0.040,-1.508) ~~ \lambda = (-0.040,1.508)$} 
I believe all the roots except for $\lambda = 0.329$ are complex conjugate roots. Can someone else please verify that those are all of the roots to this polynomial and that the ones I provided are correct, Thanks. I hope this helps out if this explanation is what you were looking for. 
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