Code for both A and X are integer.
I edit the 1st version of the code (if you happens to see t) essentially a bug correction and better generation and simplification. Second edit: fix issue with non-simple eigen-value.
n = 4;
m = 5;
p = 5;
d = randi(2*m+1,[1,n])-m-1;
C = diag(d);
while true
P = randi(2*p+1,[n,n])-p-1;
detP = round(det(P));
if detP ~= 0
break
end
end
Q = round(detP * inv(P));
A = P*C*Q;
g = 0;
for i=1:n*n
g = gcd(g,abs(A(i)));
end
A = A/g;
lambda = sort(d)*(detP/g);
I = eye(n);
X = zeros(n);
s = 0;
for k=1:n
Ak = A-lambda(k)*I;
r = rank(Ak);
[~,~,E] = qr(Ak);
[p,~] = find(E);
j1 = p(r+1:end);
j2 = p(1:r);
[~,~,E] = qr(Ak(:,j2)');
[p,~] = find(E);
i1 = p(r+1:end);
i2 = p(1:r);
Asub = Ak(i2,j2);
s = mod(s,length(j1))+1;
x = Ak(:,j2) \ Ak(:,j1(s));
y = zeros(n-r,1);
y(s) = -1;
x = round([x; y]*det(Asub));
g = 0;
for i=1:n
g = gcd(g,abs(x(i)));
end
X([j2;j1],k) = x/g;
end
D = diag(lambda);
A
X
A*X
X*D
