# Faster way to find the eigenvalues of a 4x4 real matrix?

I want to calculate the eigenvalues and eigenspaces of this matrix for self-study:

$$\frac{1}{31}\left( \begin{array}{rrr} 43 & 9 & -23 & -61\\ 16 & -19 & -10 & 22 \\ 130 & 51 & -89 &-108 \\ 36 & -4 & -7 & -59\\ \end{array}\right)$$

I tried using the normal method of finding $$det(A - \lambda I)$$, but just these massive numbers kept coming up. I also tried using different block matrix formulae, but I ended up with the same problem. I wanted to ask if there was maybe a better more efficient way of finding the eigenvalues using some trick.

I know that the trace of this matrix and therefore the sum of the eigenvalues is -4, but beyond that suggestions are appreciated. I am not looking for a full solution here.

• Suggest assuming integer eigenvalues, further guessing that one of them might be $-1.$ To confirm the guess, define $M = A +I$ and use row operations to see if $M$ is singular, i.e. reduces to a matrix with a zero row. I'm trying to get you to construct a Jordan Normal Form..... Commented Mar 30 at 17:14

Hint: It is not too difficult to compute the characteristic polynomial of your matrix $$A$$. It is given by $$\chi_A(t)=(t+1)^4.$$ Let $$J=\begin{pmatrix} -1 & 1 & 0 & 0 \cr 0 & -1 & 1 & 0 \cr 0 & 0 & -1 & 1 \cr 0 & 0 & 0 & -1 \end{pmatrix}$$ It is easy to find an invertible matrix $$S$$ such that $$SA=JS$$ by solving a system of linear equations in the entries of $$S$$. In other words, your matrix is similar to $$J$$, and you can read off all invariants.
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