Forgive me in advance if any of this is trivial. After looking at many 2x2 matrices it seems that if all of the elements in matrix are unique squared integers then the eigenvalues are irrational. So I tried to investigate this:
$\det \begin{pmatrix} \lambda -a^2 & b^2\\ c^2& \lambda -d^2\end{pmatrix}= \lambda^2 -(a^2+d^2)\lambda + (a^2d^2-c^2b^2)$
after applying the quadratic formula this gives a radical of,
$\sqrt{a^4+4b^2c^2-2a^2d^2+d^4}$
If the stated observation is true, is there a way to show that this is irrational? Furthermore it looks like on the surface that for 3x3 matrices the eigenvalues for a matrix containing all unique squared entries that the eigenvalues will also be irrational. Are either of these statements true? Is there a generalization of this for an nxn matrix?
Edit: I'm not entirely sure I derived the radical correctly, but I'd still like to have some direction on the questions above also I'd like to examine cases where the eigenvalue is not zero
Example:
\begin{pmatrix}
2^2 & 4^2\\
3^2 & 6^2
\end{pmatrix}
has eigenvalues 40 and 0.
Edit 2: still looking for rational eigenvalues of a $3×3$ have been with imposed restrictions and nonzero eigenvalues/entries.
Best Answer
The claim is not true. The matrix $$\begin{bmatrix}1^2&36^2\\5^2&26^2\end{bmatrix}$$ has eigenvalues $721$ and $-44$, which are evidently rational.