Bonus homework 2

This is yet another one of these bonus assignments. It's a quick exploration of eigenvalues and eigenvectors for some specific matrices following a certain entry-pattern. The matrices I'm interested in

are square ($n\times n$ for some positive integer $n$);
have a entry $x$ all along the diagonal, and
have entry $y$ everywhere else.

Let's just define a Python function that returns such a matrix, given $n$, $x$ and $y$ as parameters.

from sympy import *
init_printing(use_latex='mathjax')

I'll do this by

first filling the matrix with $1$ using SymPy's ones function,
then scaling that by my desired off-diagonal entry,
and then modifying the diagonal by adding some scalar multiple of the identity $n\times n$ matrix, eye($n$).

def mtrx(size, diag_entry, other_entry):
    return other_entry*ones(size) + (diag_entry-other_entry)*eye(size)

Let's try it out:

mtrx(4, 7, 19)

mtrx(10,1,5)

And so on. Your task:

Tell me, for arbitrary sise $n$, diagonal entry $x$ and off-diagonal entry $y$, the eigenvalues of the matrix matr(n,x,y), and also describe the eigenvectors.

I want the general pattern, that will of course be dependent on $n$, $x$ and $y$. I suggest you explore small cases using SymPy's eigenvals() and eigenvects() methods.