U = 
      [[.5, .7, .1, .5], [.5, -.7, -.1, .5], [.5, .1, -.7, -.5], [.5, -.1, .7, -.5]]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Integer matrices         %
% with integer eigenvalues %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Eigenvalues: 2, -6, 44, -4
U*diag([2, -6, 44, -4])*U' = 
      [[-3, 2, -2, 5], [2, -3, 5, -2], [-2, 5, 21, -22], [5, -2, -22, 21]]

% Eigenvalues: 2, -6, 44, 4
U*diag([2, -6, 44, 4])*U' = 
      [[-1, 4, -4, 3], [4, -1, 3, -4], [-4, 3, 23, -20], [3, -4, -20, 23]]

% Eigenvalues: 2, 6, -44, -4
U*diag([2, 6, -44, -4])*U' = 
      [[2, -3, 5, -2], [-3, 2, -2, 5], [5, -2, -22, 21], [-2, 5, 21, -22]]

% Eigenvalues: 2, 6, -44, 4
U*diag([2, 6, -44, 4])*U' = 
      [[4, -1, 3, -4], [-1, 4, -4, 3], [3, -4, -20, 23], [-4, 3, 23, -20]]

% Eigenvalues: 1, -5, 45, -1
U*diag([1, -5, 45, -1])*U' = 
      [[-2, 2, -3, 4], [2, -2, 4, -3], [-3, 4, 22, -22], [4, -3, -22, 22]]

% Eigenvalues: 1, 5, -45, -1
U*diag([1, 5, -45, -1])*U' = 
      [[2, -2, 4, -3], [-2, 2, -3, 4], [4, -3, -22, 22], [-3, 4, 22, -22]]

% Eigenvalues: 1, -5, 45, 3
U*diag([1, -5, 45, 3])*U' = 
      [[-1, 3, -4, 3], [3, -1, 3, -4], [-4, 3, 23, -21], [3, -4, -21, 23]]

% Eigenvalues: 1, 5, -45, 3
U*diag([1, 5, -45, 3])*U' = 
      [[3, -1, 3, -4], [-1, 3, -4, 3], [3, -4, -21, 23], [-4, 3, 23, -21]]

% Eigenvalues: 3, -5, 45, -3
U*diag([3, -5, 45, -3])*U' = 
      [[-2, 2, -2, 5], [2, -2, 5, -2], [-2, 5, 22, -22], [5, -2, -22, 22]]

% Eigenvalues: 3, 5, -45, -3
U*diag([3, 5, -45, -3])*U' = 
      [[2, -2, 5, -2], [-2, 2, -2, 5], [5, -2, -22, 22], [-2, 5, 22, -22]]

% Eigenvalues: 1, 5, -45, -5
U*diag([1, 5, -45, -5])*U' = 
      [[1, -3, 5, -2], [-3, 1, -2, 5], [5, -2, -23, 21], [-2, 5, 21, -23]]

% Eigenvalues: 2, -4, 46, 4
U*diag([2, -4, 46, 4])*U' = 
      [[0, 3, -4, 3], [3, 0, 3, -4], [-4, 3, 24, -21], [3, -4, -21, 24]]

% Eigenvalues: 2, 4, -46, -4
U*diag([2, 4, -46, -4])*U' = 
      [[1, -2, 5, -2], [-2, 1, -2, 5], [5, -2, -23, 22], [-2, 5, 22, -23]]

% Eigenvalues: 4, -4, 46, -2
U*diag([4, -4, 46, -2])*U' = 
      [[-1, 2, -2, 5], [2, -1, 5, -2], [-2, 5, 23, -22], [5, -2, -22, 23]]

% Eigenvalues: 6, 4, -46, -4
U*diag([6, 4, -46, -4])*U' = 
      [[2, -1, 6, -1], [-1, 2, -1, 6], [6, -1, -22, 23], [-1, 6, 23, -22]]

% Eigenvalues: 2, -4, 46, -8
U*diag([2, -4, 46, -8])*U' = 
      [[-3, 0, -1, 6], [0, -3, 6, -1], [-1, 6, 21, -24], [6, -1, -24, 21]]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Matrices with at most 1 digit %
% beyond the decimal point      %
% with integer eigenvalues      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Eigenvalues: 1, 2, -3, -2
U*diag([[10, 20, -30, -20]])*U'/10 = 
      [[.7, -1.2, 1.1, .4], [-1.2, .7, .4, 1.1], [1.1, .4, -1.7, 1.2], [.4, 1.1, 1.2, -1.7]]

% Eigenvalues: 2, -2, 3, -1
U*diag([[20, -20, 30, -10]])*U'/10 = 
      [[-.7, 1.2, .4, 1.1], [1.2, -.7, 1.1, .4], [.4, 1.1, 1.7, -1.2], [1.1, .4, -1.2, 1.7]]

% Eigenvalues: 1, -2, 3, 2
U*diag([[10, -20, 30, 20]])*U'/10 = 
      [[-.2, 1.7, -.6, .1], [1.7, -.2, .1, -.6], [-.6, .1, 2.2, -.7], [.1, -.6, -.7, 2.2]]

% Eigenvalues: 2, 1, -4, -1
U*diag([[20, 10, -40, -10]])*U'/10 = 
      [[.7, -.2, 1.1, .4], [-.2, .7, .4, 1.1], [1.1, .4, -1.7, 2.2], [.4, 1.1, 2.2, -1.7]]

% Eigenvalues: 1, -1, 4, -2
U*diag([[10, -10, 40, -20]])*U'/10 = 
      [[-.7, .2, .4, 1.1], [.2, -.7, 1.1, .4], [.4, 1.1, 1.7, -2.2], [1.1, .4, -2.2, 1.7]]

% Eigenvalues: 1, -1, 4, 2
U*diag([[10, -10, 40, 20]])*U'/10 = 
      [[.3, 1.2, -.6, .1], [1.2, .3, .1, -.6], [-.6, .1, 2.7, -1.2], [.1, -.6, -1.2, 2.7]]

% Eigenvalues: 3, 2, -3, -2
U*diag([[30, 20, -30, -20]])*U'/10 = 
      [[1.2, -.7, 1.6, .9], [-.7, 1.2, .9, 1.6], [1.6, .9, -1.2, 1.7], [.9, 1.6, 1.7, -1.2]]

% Eigenvalues: 3, 1, -4, -2
U*diag([[30, 10, -40, -20]])*U'/10 = 
      [[.7, -.2, 1.6, .9], [-.2, .7, .9, 1.6], [1.6, .9, -1.7, 2.2], [.9, 1.6, 2.2, -1.7]]

% Eigenvalues: 2, -1, 4, -3
U*diag([[20, -10, 40, -30]])*U'/10 = 
      [[-.7, .2, .9, 1.6], [.2, -.7, 1.6, .9], [.9, 1.6, 1.7, -2.2], [1.6, .9, -2.2, 1.7]]

% Eigenvalues: 2, 1, 6, -1
U*diag([[20, 10, 60, -10]])*U'/10 = 
      [[.8, -.3, .4, 1.1], [-.3, .8, 1.1, .4], [.4, 1.1, 3.2, -2.7], [1.1, .4, -2.7, 3.2]]

% Eigenvalues: 1, -1, -6, -2
U*diag([[10, -10, -60, -20]])*U'/10 = 
      [[-.8, .3, 1.1, .4], [.3, -.8, .4, 1.1], [1.1, .4, -3.2, 2.7], [.4, 1.1, 2.7, -3.2]]

% Eigenvalues: 1, -1, -6, 2
U*diag([[10, -10, -60, 20]])*U'/10 = 
      [[.2, 1.3, .1, -.6], [1.3, .2, -.6, .1], [.1, -.6, -2.2, 3.7], [-.6, .1, 3.7, -2.2]]

% Eigenvalues: 3, 1, 6, -2
U*diag([[30, 10, 60, -20]])*U'/10 = 
      [[.8, -.3, .9, 1.6], [-.3, .8, 1.6, .9], [.9, 1.6, 3.2, -2.7], [1.6, .9, -2.7, 3.2]]

% Eigenvalues: 2, -1, -6, -3
U*diag([[20, -10, -60, -30]])*U'/10 = 
      [[-.8, .3, 1.6, .9], [.3, -.8, .9, 1.6], [1.6, .9, -3.2, 2.7], [.9, 1.6, 2.7, -3.2]]

% Eigenvalues: 2, 1, 6, 3
U*diag([[20, 10, 60, 30]])*U'/10 = 
      [[1.8, .7, -.6, .1], [.7, 1.8, .1, -.6], [-.6, .1, 4.2, -1.7], [.1, -.6, -1.7, 4.2]]

% Eigenvalues: 1, -1, -6, -4
U*diag([[10, -10, -60, -40]])*U'/10 = 
      [[-1.3, -.2, 1.6, .9], [-.2, -1.3, .9, 1.6], [1.6, .9, -3.7, 2.2], [.9, 1.6, 2.2, -3.7]]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Matrices with at most 2 digits %
% beyond the decimal point       %
% with integer eigenvalues       %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Eigenvalues: 1, -2, -1, 2
U*diag([[100, -200, -100, 200]])*U'/100 = 
      [[-.24, 1.74, -.32, -.18], [1.74, -.24, -.18, -.32], [-.32, -.18, .24, 1.26], [-.18, -.32, 1.26, .24]]