U = 
      [[.1, .3, .3, .9], [.3, -.1, -.9, .3], [.3, .9, -.1, -.3], [.9, -.3, .3, -.1]]

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

% Eigenvalues: 45, -5, -45, 5
U*diag([45, -5, -45, 5])*U' = 
      [[0, 15, 0, 0], [15, -32, 0, 24], [0, 0, 0, 15], [0, 24, 15, 32]]

% Eigenvalues: 44, -6, -46, 4
U*diag([44, -6, -46, 4])*U' = 
      [[-1, 15, 0, 0], [15, -33, 0, 24], [0, 0, -1, 15], [0, 24, 15, 31]]

% Eigenvalues: 46, -4, -44, 6
U*diag([46, -4, -44, 6])*U' = 
      [[1, 15, 0, 0], [15, -31, 0, 24], [0, 0, 1, 15], [0, 24, 15, 33]]

% Eigenvalues: 43, -7, -47, 3
U*diag([43, -7, -47, 3])*U' = 
      [[-2, 15, 0, 0], [15, -34, 0, 24], [0, 0, -2, 15], [0, 24, 15, 30]]

% Eigenvalues: 47, -3, -43, 7
U*diag([47, -3, -43, 7])*U' = 
      [[2, 15, 0, 0], [15, -30, 0, 24], [0, 0, 2, 15], [0, 24, 15, 34]]

% Eigenvalues: 42, -8, -48, 2
U*diag([42, -8, -48, 2])*U' = 
      [[-3, 15, 0, 0], [15, -35, 0, 24], [0, 0, -3, 15], [0, 24, 15, 29]]

% Eigenvalues: 41, -9, 51, 1
U*diag([41, -9, 51, 1])*U' = 
      [[5, -12, -3, 9], [-12, 45, 9, -3], [-3, 9, -3, 12], [9, -3, 12, 37]]

% Eigenvalues: 42, -8, 52, 2
U*diag([42, -8, 52, 2])*U' = 
      [[6, -12, -3, 9], [-12, 46, 9, -3], [-3, 9, -2, 12], [9, -3, 12, 38]]

% Eigenvalues: 43, -7, 53, 3
U*diag([43, -7, 53, 3])*U' = 
      [[7, -12, -3, 9], [-12, 47, 9, -3], [-3, 9, -1, 12], [9, -3, 12, 39]]

% Eigenvalues: 44, -6, 54, 4
U*diag([44, -6, 54, 4])*U' = 
      [[8, -12, -3, 9], [-12, 48, 9, -3], [-3, 9, 0, 12], [9, -3, 12, 40]]

% Eigenvalues: 45, -5, 55, 5
U*diag([45, -5, 55, 5])*U' = 
      [[9, -12, -3, 9], [-12, 49, 9, -3], [-3, 9, 1, 12], [9, -3, 12, 41]]

% Eigenvalues: 46, -4, 56, 6
U*diag([46, -4, 56, 6])*U' = 
      [[10, -12, -3, 9], [-12, 50, 9, -3], [-3, 9, 2, 12], [9, -3, 12, 42]]

% Eigenvalues: 90, -10, -90, 10
U*diag([90, -10, -90, 10])*U' = 
      [[0, 30, 0, 0], [30, -64, 0, 48], [0, 0, 0, 30], [0, 48, 30, 64]]

% Eigenvalues: 89, -11, -91, 9
U*diag([89, -11, -91, 9])*U' = 
      [[-1, 30, 0, 0], [30, -65, 0, 48], [0, 0, -1, 30], [0, 48, 30, 63]]

% Eigenvalues: 88, -12, -92, 8
U*diag([88, -12, -92, 8])*U' = 
      [[-2, 30, 0, 0], [30, -66, 0, 48], [0, 0, -2, 30], [0, 48, 30, 62]]

% Eigenvalues: 87, -13, -93, 7
U*diag([87, -13, -93, 7])*U' = 
      [[-3, 30, 0, 0], [30, -67, 0, 48], [0, 0, -3, 30], [0, 48, 30, 61]]

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

% Eigenvalues: 3, -2, -3, 2
U*diag([[30, -20, -30, 20]])*U'/10 = 
      [[1.2, 1.5, -.9, 0], [1.5, -2, 0, 1.5], [-.9, 0, -1.2, 1.5], [0, 1.5, 1.5, 2]]

% Eigenvalues: 3, -2, 4, -1
U*diag([[30, -20, 40, -10]])*U'/10 = 
      [[-.6, -1.2, -.3, .9], [-1.2, 3.4, .9, -.3], [-.3, .9, -1.4, 1.2], [.9, -.3, 1.2, 2.6]]

% Eigenvalues: 3, -2, -4, 1
U*diag([[30, -20, -40, 10]])*U'/10 = 
      [[.3, 1.5, -.6, 0], [1.5, -2.9, 0, 1.8], [-.6, 0, -1.3, 1.5], [0, 1.8, 1.5, 1.9]]

% Eigenvalues: 4, -1, -3, 2
U*diag([[40, -10, -30, 20]])*U'/10 = 
      [[1.3, 1.5, -.6, 0], [1.5, -1.9, 0, 1.8], [-.6, 0, -.3, 1.5], [0, 1.8, 1.5, 2.9]]

% Eigenvalues: 4, -1, -4, 1
U*diag([[40, -10, -40, 10]])*U'/10 = 
      [[.4, 1.5, -.3, 0], [1.5, -2.8, 0, 2.1], [-.3, 0, -.4, 1.5], [0, 2.1, 1.5, 2.8]]

% Eigenvalues: 3, -2, -6, -1
U*diag([[30, -20, -60, -10]])*U'/10 = 
      [[-1.5, 1.5, 0, 0], [1.5, -4.7, 0, 2.4], [0, 0, -1.5, 1.5], [0, 2.4, 1.5, 1.7]]

% Eigenvalues: 6, 1, 2, -3
U*diag([[60, 10, 20, -30]])*U'/10 = 
      [[-2.1, -1.2, 1.2, .9], [-1.2, 1.9, .9, 1.2], [1.2, .9, 1.1, 1.2], [.9, 1.2, 1.2, 5.1]]

% Eigenvalues: 6, 1, 3, -2
U*diag([[60, 10, 30, -20]])*U'/10 = 
      [[-1.2, -1.2, .9, .9], [-1.2, 2.8, .9, .9], [.9, .9, 1.2, 1.2], [.9, .9, 1.2, 5.2]]

% Eigenvalues: 4, -1, 6, 1
U*diag([[40, -10, 60, 10]])*U'/10 = 
      [[1.3, -1.2, -.6, .9], [-1.2, 5.3, .9, -.6], [-.6, .9, -.3, 1.2], [.9, -.6, 1.2, 3.7]]

% Eigenvalues: 7, 2, 3, -2
U*diag([[70, 20, 30, -20]])*U'/10 = 
      [[-1.1, -1.2, 1.2, .9], [-1.2, 2.9, .9, 1.2], [1.2, .9, 2.1, 1.2], [.9, 1.2, 1.2, 6.1]]

% Eigenvalues: 4, -1, 7, 2
U*diag([[40, -10, 70, 20]])*U'/10 = 
      [[2.2, -1.2, -.9, .9], [-1.2, 6.2, .9, -.9], [-.9, .9, -.2, 1.2], [.9, -.9, 1.2, 3.8]]

% Eigenvalues: 6, 1, -6, -1
U*diag([[60, 10, -60, -10]])*U'/10 = 
      [[-1.2, 1.5, .9, 0], [1.5, -4.4, 0, 3.3], [.9, 0, 1.2, 1.5], [0, 3.3, 1.5, 4.4]]

% Eigenvalues: 4, -1, 8, 3
U*diag([[40, -10, 80, 30]])*U'/10 = 
      [[3.1, -1.2, -1.2, .9], [-1.2, 7.1, .9, -1.2], [-1.2, .9, -.1, 1.2], [.9, -1.2, 1.2, 3.9]]

% Eigenvalues: 7, 2, -1, -6
U*diag([[70, 20, -10, -60]])*U'/10 = 
      [[-4.7, -1.2, 2.4, .9], [-1.2, -.7, .9, 2.4], [2.4, .9, 1.7, 1.2], [.9, 2.4, 1.2, 5.7]]

% Eigenvalues: 7, 2, -7, -2
U*diag([[70, 20, -70, -20]])*U'/10 = 
      [[-2, 1.5, 1.5, 0], [1.5, -5.2, 0, 3.9], [1.5, 0, 2, 1.5], [0, 3.9, 1.5, 5.2]]

% Eigenvalues: 6, 1, -8, -3
U*diag([[60, 10, -80, -30]])*U'/10 = 
      [[-3, 1.5, 1.5, 0], [1.5, -6.2, 0, 3.9], [1.5, 0, 1, 1.5], [0, 3.9, 1.5, 4.2]]

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

% Eigenvalues: 1, 2, -2, -1
U*diag([[100, 200, -200, -100]])*U'/100 = 
      [[-.8, .24, .9, -.18], [.24, -1.6, -.18, .9], [.9, -.18, 1.6, -.24], [-.18, .9, -.24, .8]]

% Eigenvalues: 1, -1, 3, 2
U*diag([[100, -100, 300, 200]])*U'/100 = 
      [[1.81, -.21, -.87, .27], [-.21, 2.69, .27, -.63], [-.87, .27, -.51, .51], [.27, -.63, .51, 1.01]]

% Eigenvalues: 1, -1, -3, 2
U*diag([[100, -100, -300, 200]])*U'/100 = 
      [[1.27, 1.41, -.69, -.27], [1.41, -2.17, -.27, .99], [-.69, -.27, -.57, .69], [-.27, .99, .69, .47]]

% Eigenvalues: 2, 1, -3, -2
U*diag([[200, 100, -300, -200]])*U'/100 = 
      [[-1.78, .3, .96, 0], [.3, -2.42, 0, 1.44], [.96, 0, .78, .3], [0, 1.44, .3, 1.42]]

% Eigenvalues: 4, 2, 1, -1
U*diag([[400, 200, 100, -100]])*U'/100 = 
      [[-.5, -.48, .9, .36], [-.48, 1.1, .36, .9], [.9, .36, 1.9, .48], [.36, .9, .48, 3.5]]

% Eigenvalues: 1, -1, -4, 2
U*diag([[100, -100, -400, 200]])*U'/100 = 
      [[1.18, 1.68, -.66, -.36], [1.68, -2.98, -.36, 1.26], [-.66, -.36, -.58, .72], [-.36, 1.26, .72, .38]]

% Eigenvalues: 4, -2, 1, -1
U*diag([[400, -200, 100, -100]])*U'/100 = 
      [[-.86, -.36, -.18, .72], [-.36, 1.06, .72, .78], [-.18, .72, -1.34, 1.56], [.72, .78, 1.56, 3.14]]

% Eigenvalues: 3, 1, 2, -3
U*diag([[300, 100, 200, -300]])*U'/100 = 
      [[-2.13, -1.29, 1.11, .63], [-1.29, 1.63, .63, .39], [1.11, .63, .83, .39], [.63, .39, .39, 2.67]]

% Eigenvalues: 2, 1, -4, -2
U*diag([[200, 100, -400, -200]])*U'/100 = 
      [[-1.87, .57, .99, -.09], [.57, -3.23, -.09, 1.71], [.99, -.09, .77, .33], [-.09, 1.71, .33, 1.33]]

% Eigenvalues: 1, -2, 2, 4
U*diag([[100, -200, 200, 400]])*U'/100 = 
      [[3.25, .63, -1.65, .09], [.63, 2.05, .09, -.45], [-1.65, .09, -1.15, .87], [.09, -.45, .87, .85]]

% Eigenvalues: 4, -2, 2, -1
U*diag([[400, -200, 200, -100]])*U'/100 = 
      [[-.77, -.63, -.21, .81], [-.63, 1.87, .81, .51], [-.21, .81, -1.33, 1.53], [.81, .51, 1.53, 3.23]]

% Eigenvalues: 4, 1, 3, -1
U*diag([[400, 100, 300, -100]])*U'/100 = 
      [[-.41, -.99, .57, .63], [-.99, 2.71, .63, .33], [.57, .63, 1.11, .69], [.63, .33, .69, 3.59]]

% Eigenvalues: 4, 2, 3, -1
U*diag([[400, 200, 300, -100]])*U'/100 = 
      [[-.32, -1.02, .84, .54], [-1.02, 2.72, .54, .36], [.84, .54, 1.92, .42], [.54, .36, .42, 3.68]]

% Eigenvalues: 4, 1, -2, -3
U*diag([[400, 100, -200, -300]])*U'/100 = 
      [[-2.48, -.18, 1.26, .36], [-.18, -1.52, .36, 1.74], [1.26, .36, .88, .78], [.36, 1.74, .78, 3.12]]

% Eigenvalues: 1, -2, 3, 4
U*diag([[100, -200, 300, 400]])*U'/100 = 
      [[3.34, .36, -1.68, .18], [.36, 2.86, .18, -.72], [-1.68, .18, -1.14, .84], [.18, -.72, .84, .94]]

% Eigenvalues: 4, -2, 3, -1
U*diag([[400, -200, 300, -100]])*U'/100 = 
      [[-.68, -.9, -.24, .9], [-.9, 2.68, .9, .24], [-.24, .9, -1.32, 1.5], [.9, .24, 1.5, 3.32]]

U = 
      [[.1, .3, .3, .9], [.3, -.1, -.9, .3], [.3, .9, -.1, -.3], [.9, -.3, .3, -.1]]

U' = 
      [[.1, .3, .3, .9], [.3, -.1, .9, -.3], [.3, -.9, -.1, .3], [.9, .3, -.3, -.1]]

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

% Eigenvalues: 45, -5, -45, 5
U*diag([45, -5, -45, 5])*U' = 
      [[0, 15, 0, 0], [15, -32, 0, 24], [0, 0, 0, 15], [0, 24, 15, 32]]

% Eigenvalues: 44, -6, -46, 4
U*diag([44, -6, -46, 4])*U' = 
      [[-1, 15, 0, 0], [15, -33, 0, 24], [0, 0, -1, 15], [0, 24, 15, 31]]

% Eigenvalues: 46, -4, -44, 6
U*diag([46, -4, -44, 6])*U' = 
      [[1, 15, 0, 0], [15, -31, 0, 24], [0, 0, 1, 15], [0, 24, 15, 33]]

% Eigenvalues: 43, -7, -47, 3
U*diag([43, -7, -47, 3])*U' = 
      [[-2, 15, 0, 0], [15, -34, 0, 24], [0, 0, -2, 15], [0, 24, 15, 30]]

% Eigenvalues: 47, -3, -43, 7
U*diag([47, -3, -43, 7])*U' = 
      [[2, 15, 0, 0], [15, -30, 0, 24], [0, 0, 2, 15], [0, 24, 15, 34]]

% Eigenvalues: 42, -8, -48, 2
U*diag([42, -8, -48, 2])*U' = 
      [[-3, 15, 0, 0], [15, -35, 0, 24], [0, 0, -3, 15], [0, 24, 15, 29]]

% Eigenvalues: 41, -9, 51, 1
U*diag([41, -9, 51, 1])*U' = 
      [[5, -12, -3, 9], [-12, 45, 9, -3], [-3, 9, -3, 12], [9, -3, 12, 37]]

% Eigenvalues: 42, -8, 52, 2
U*diag([42, -8, 52, 2])*U' = 
      [[6, -12, -3, 9], [-12, 46, 9, -3], [-3, 9, -2, 12], [9, -3, 12, 38]]

% Eigenvalues: 43, -7, 53, 3
U*diag([43, -7, 53, 3])*U' = 
      [[7, -12, -3, 9], [-12, 47, 9, -3], [-3, 9, -1, 12], [9, -3, 12, 39]]

% Eigenvalues: 44, -6, 54, 4
U*diag([44, -6, 54, 4])*U' = 
      [[8, -12, -3, 9], [-12, 48, 9, -3], [-3, 9, 0, 12], [9, -3, 12, 40]]

% Eigenvalues: 45, -5, 55, 5
U*diag([45, -5, 55, 5])*U' = 
      [[9, -12, -3, 9], [-12, 49, 9, -3], [-3, 9, 1, 12], [9, -3, 12, 41]]

% Eigenvalues: 46, -4, 56, 6
U*diag([46, -4, 56, 6])*U' = 
      [[10, -12, -3, 9], [-12, 50, 9, -3], [-3, 9, 2, 12], [9, -3, 12, 42]]

% Eigenvalues: 90, -10, -90, 10
U*diag([90, -10, -90, 10])*U' = 
      [[0, 30, 0, 0], [30, -64, 0, 48], [0, 0, 0, 30], [0, 48, 30, 64]]

% Eigenvalues: 89, -11, -91, 9
U*diag([89, -11, -91, 9])*U' = 
      [[-1, 30, 0, 0], [30, -65, 0, 48], [0, 0, -1, 30], [0, 48, 30, 63]]

% Eigenvalues: 88, -12, -92, 8
U*diag([88, -12, -92, 8])*U' = 
      [[-2, 30, 0, 0], [30, -66, 0, 48], [0, 0, -2, 30], [0, 48, 30, 62]]

% Eigenvalues: 87, -13, -93, 7
U*diag([87, -13, -93, 7])*U' = 
      [[-3, 30, 0, 0], [30, -67, 0, 48], [0, 0, -3, 30], [0, 48, 30, 61]]

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

% Eigenvalues: 3, -2, -3, 2
U*diag([[30, -20, -30, 20]])*U'/10 = 
      [[1.2, 1.5, -.9, 0], [1.5, -2, 0, 1.5], [-.9, 0, -1.2, 1.5], [0, 1.5, 1.5, 2]]

% Eigenvalues: 3, -2, 4, -1
U*diag([[30, -20, 40, -10]])*U'/10 = 
      [[-.6, -1.2, -.3, .9], [-1.2, 3.4, .9, -.3], [-.3, .9, -1.4, 1.2], [.9, -.3, 1.2, 2.6]]

% Eigenvalues: 3, -2, -4, 1
U*diag([[30, -20, -40, 10]])*U'/10 = 
      [[.3, 1.5, -.6, 0], [1.5, -2.9, 0, 1.8], [-.6, 0, -1.3, 1.5], [0, 1.8, 1.5, 1.9]]

% Eigenvalues: 4, -1, -3, 2
U*diag([[40, -10, -30, 20]])*U'/10 = 
      [[1.3, 1.5, -.6, 0], [1.5, -1.9, 0, 1.8], [-.6, 0, -.3, 1.5], [0, 1.8, 1.5, 2.9]]

% Eigenvalues: 4, -1, -4, 1
U*diag([[40, -10, -40, 10]])*U'/10 = 
      [[.4, 1.5, -.3, 0], [1.5, -2.8, 0, 2.1], [-.3, 0, -.4, 1.5], [0, 2.1, 1.5, 2.8]]

% Eigenvalues: 3, -2, -6, -1
U*diag([[30, -20, -60, -10]])*U'/10 = 
      [[-1.5, 1.5, 0, 0], [1.5, -4.7, 0, 2.4], [0, 0, -1.5, 1.5], [0, 2.4, 1.5, 1.7]]

% Eigenvalues: 6, 1, 2, -3
U*diag([[60, 10, 20, -30]])*U'/10 = 
      [[-2.1, -1.2, 1.2, .9], [-1.2, 1.9, .9, 1.2], [1.2, .9, 1.1, 1.2], [.9, 1.2, 1.2, 5.1]]

% Eigenvalues: 6, 1, 3, -2
U*diag([[60, 10, 30, -20]])*U'/10 = 
      [[-1.2, -1.2, .9, .9], [-1.2, 2.8, .9, .9], [.9, .9, 1.2, 1.2], [.9, .9, 1.2, 5.2]]

% Eigenvalues: 4, -1, 6, 1
U*diag([[40, -10, 60, 10]])*U'/10 = 
      [[1.3, -1.2, -.6, .9], [-1.2, 5.3, .9, -.6], [-.6, .9, -.3, 1.2], [.9, -.6, 1.2, 3.7]]

% Eigenvalues: 7, 2, 3, -2
U*diag([[70, 20, 30, -20]])*U'/10 = 
      [[-1.1, -1.2, 1.2, .9], [-1.2, 2.9, .9, 1.2], [1.2, .9, 2.1, 1.2], [.9, 1.2, 1.2, 6.1]]

% Eigenvalues: 4, -1, 7, 2
U*diag([[40, -10, 70, 20]])*U'/10 = 
      [[2.2, -1.2, -.9, .9], [-1.2, 6.2, .9, -.9], [-.9, .9, -.2, 1.2], [.9, -.9, 1.2, 3.8]]

% Eigenvalues: 6, 1, -6, -1
U*diag([[60, 10, -60, -10]])*U'/10 = 
      [[-1.2, 1.5, .9, 0], [1.5, -4.4, 0, 3.3], [.9, 0, 1.2, 1.5], [0, 3.3, 1.5, 4.4]]

% Eigenvalues: 4, -1, 8, 3
U*diag([[40, -10, 80, 30]])*U'/10 = 
      [[3.1, -1.2, -1.2, .9], [-1.2, 7.1, .9, -1.2], [-1.2, .9, -.1, 1.2], [.9, -1.2, 1.2, 3.9]]

% Eigenvalues: 7, 2, -1, -6
U*diag([[70, 20, -10, -60]])*U'/10 = 
      [[-4.7, -1.2, 2.4, .9], [-1.2, -.7, .9, 2.4], [2.4, .9, 1.7, 1.2], [.9, 2.4, 1.2, 5.7]]

% Eigenvalues: 7, 2, -7, -2
U*diag([[70, 20, -70, -20]])*U'/10 = 
      [[-2, 1.5, 1.5, 0], [1.5, -5.2, 0, 3.9], [1.5, 0, 2, 1.5], [0, 3.9, 1.5, 5.2]]

% Eigenvalues: 6, 1, -8, -3
U*diag([[60, 10, -80, -30]])*U'/10 = 
      [[-3, 1.5, 1.5, 0], [1.5, -6.2, 0, 3.9], [1.5, 0, 1, 1.5], [0, 3.9, 1.5, 4.2]]

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

% Eigenvalues: 1, 2, -2, -1
U*diag([[100, 200, -200, -100]])*U'/100 = 
      [[-.8, .24, .9, -.18], [.24, -1.6, -.18, .9], [.9, -.18, 1.6, -.24], [-.18, .9, -.24, .8]]

% Eigenvalues: 1, -1, 3, 2
U*diag([[100, -100, 300, 200]])*U'/100 = 
      [[1.81, -.21, -.87, .27], [-.21, 2.69, .27, -.63], [-.87, .27, -.51, .51], [.27, -.63, .51, 1.01]]

% Eigenvalues: 1, -1, -3, 2
U*diag([[100, -100, -300, 200]])*U'/100 = 
      [[1.27, 1.41, -.69, -.27], [1.41, -2.17, -.27, .99], [-.69, -.27, -.57, .69], [-.27, .99, .69, .47]]

% Eigenvalues: 2, 1, -3, -2
U*diag([[200, 100, -300, -200]])*U'/100 = 
      [[-1.78, .3, .96, 0], [.3, -2.42, 0, 1.44], [.96, 0, .78, .3], [0, 1.44, .3, 1.42]]

% Eigenvalues: 4, 2, 1, -1
U*diag([[400, 200, 100, -100]])*U'/100 = 
      [[-.5, -.48, .9, .36], [-.48, 1.1, .36, .9], [.9, .36, 1.9, .48], [.36, .9, .48, 3.5]]

% Eigenvalues: 1, -1, -4, 2
U*diag([[100, -100, -400, 200]])*U'/100 = 
      [[1.18, 1.68, -.66, -.36], [1.68, -2.98, -.36, 1.26], [-.66, -.36, -.58, .72], [-.36, 1.26, .72, .38]]

% Eigenvalues: 4, -2, 1, -1
U*diag([[400, -200, 100, -100]])*U'/100 = 
      [[-.86, -.36, -.18, .72], [-.36, 1.06, .72, .78], [-.18, .72, -1.34, 1.56], [.72, .78, 1.56, 3.14]]

% Eigenvalues: 3, 1, 2, -3
U*diag([[300, 100, 200, -300]])*U'/100 = 
      [[-2.13, -1.29, 1.11, .63], [-1.29, 1.63, .63, .39], [1.11, .63, .83, .39], [.63, .39, .39, 2.67]]

% Eigenvalues: 2, 1, -4, -2
U*diag([[200, 100, -400, -200]])*U'/100 = 
      [[-1.87, .57, .99, -.09], [.57, -3.23, -.09, 1.71], [.99, -.09, .77, .33], [-.09, 1.71, .33, 1.33]]

% Eigenvalues: 1, -2, 2, 4
U*diag([[100, -200, 200, 400]])*U'/100 = 
      [[3.25, .63, -1.65, .09], [.63, 2.05, .09, -.45], [-1.65, .09, -1.15, .87], [.09, -.45, .87, .85]]

% Eigenvalues: 4, -2, 2, -1
U*diag([[400, -200, 200, -100]])*U'/100 = 
      [[-.77, -.63, -.21, .81], [-.63, 1.87, .81, .51], [-.21, .81, -1.33, 1.53], [.81, .51, 1.53, 3.23]]

% Eigenvalues: 4, 1, 3, -1
U*diag([[400, 100, 300, -100]])*U'/100 = 
      [[-.41, -.99, .57, .63], [-.99, 2.71, .63, .33], [.57, .63, 1.11, .69], [.63, .33, .69, 3.59]]

% Eigenvalues: 4, 2, 3, -1
U*diag([[400, 200, 300, -100]])*U'/100 = 
      [[-.32, -1.02, .84, .54], [-1.02, 2.72, .54, .36], [.84, .54, 1.92, .42], [.54, .36, .42, 3.68]]

% Eigenvalues: 4, 1, -2, -3
U*diag([[400, 100, -200, -300]])*U'/100 = 
      [[-2.48, -.18, 1.26, .36], [-.18, -1.52, .36, 1.74], [1.26, .36, .88, .78], [.36, 1.74, .78, 3.12]]

% Eigenvalues: 1, -2, 3, 4
U*diag([[100, -200, 300, 400]])*U'/100 = 
      [[3.34, .36, -1.68, .18], [.36, 2.86, .18, -.72], [-1.68, .18, -1.14, .84], [.18, -.72, .84, .94]]

% Eigenvalues: 4, -2, 3, -1
U*diag([[400, -200, 300, -100]])*U'/100 = 
      [[-.68, -.9, -.24, .9], [-.9, 2.68, .9, .24], [-.24, .9, -1.32, 1.5], [.9, .24, 1.5, 3.32]]