U =
      [[1, 0,    0],
       [0, 0.6,  0.8],
       [0, 0.8, -0.6]]

V =
      [[0.5,  0.5,  0.5,  0.5],
       [0.5, -0.5, -0.5,  0.5],
       [0.5,  0.5, -0.5, -0.5],
       [0.5, -0.5,  0.5, -0.5]]

%%%%%%%%%%%%%%%%%%%%%%%%%
% Matrices with integer %
%    singular values    %
%%%%%%%%%%%%%%%%%%%%%%%%%

% Singular values: 2, 2, 1
U*diag([2, 2, 1])*V'
      [[1,    1,   1,    1],
       [1,   -1,   0.2, -0.2],
       [0.5, -0.5, 1.1, -1.1]]

% Singular values: 3, 2, 1
U*diag([3, 2, 1])*V'
      [[1.5,  1.5, 1.5,  1.5],
       [1,   -1,   0.2, -0.2],
       [0.5, -0.5, 1.1, -1.1]]

% Singular values: 3, 3, 1
U*diag([3, 3, 1])*V'
      [[1.5,  1.5, 1.5,  1.5],
       [1.3, -1.3, 0.5, -0.5],
       [0.9, -0.9, 1.5, -1.5]]

% Singular values: 3, 3, 2
U*diag([3, 3, 2])*V'
      [[1.5,  1.5, 1.5,  1.5],
       [1.7, -1.7, 0.1, -0.1],
       [0.6, -0.6, 1.8, -1.8]]

% Singular values: 4, 2, 1
U*diag([4, 2, 1])*V'
      [[2,    2,   2,    2],
       [1,   -1,   0.2, -0.2],
       [0.5, -0.5, 1.1, -1.1]]

% Singular values: 4, 3, 1
U*diag([4, 3, 1])*V'
      [[2,    2,   2,    2],
       [1.3, -1.3, 0.5, -0.5],
       [0.9, -0.9, 1.5, -1.5]]

% Singular values: 4, 3, 2
U*diag([4, 3, 2])*V'
      [[2, 2, 2, 2],
       [1.7, -1.7, 0.1, -0.1],
       [0.6, -0.6, 1.8, -1.8]]

% Singular values: 4, 4, 1
U*diag([4, 4, 1])*V'
      [[2, 2, 2, 2],
       [1.6, -1.6, 0.8, -0.8],
       [1.3, -1.3, 1.9, -1.9]]

% Singular values: 4, 4, 2
U*diag([4, 4, 2])*V'
      [[2, 2, 2, 2],
       [2, -2, 0.4, -0.4],
       [1, -1, 2.2, -2.2]]

% Singular values: 4, 4, 3
U*diag([4, 4, 3])*V'
      [[2, 2, 2, 2],
       [2.4, -2.4, 0, 0],
       [0.7, -0.7, 2.5, -2.5]]

% Singular values: 5, 2, 1
U*diag([5, 2, 1])*V'
      [[2.5, 2.5, 2.5, 2.5],
       [1, -1, 0.2, -0.2],
       [0.5, -0.5, 1.1, -1.1]]

% Singular values: 5, 3, 1
U*diag([5, 3, 1])*V'
      [[2.5, 2.5, 2.5, 2.5],
       [1.3, -1.3, 0.5, -0.5],
       [0.9, -0.9, 1.5, -1.5]]

% Singular values: 5, 3, 2
U*diag([5, 3, 2])*V'
      [[2.5, 2.5, 2.5, 2.5],
       [1.7, -1.7, 0.1, -0.1],
       [0.6, -0.6, 1.8, -1.8]]

% Singular values: 5, 4, 1
U*diag([5, 4, 1])*V'
      [[2.5, 2.5, 2.5, 2.5],
       [1.6, -1.6, 0.8, -0.8],
       [1.3, -1.3, 1.9, -1.9]]

% Singular values: 5, 4, 2
U*diag([5, 4, 2])*V'
      [[2.5, 2.5, 2.5, 2.5],
       [2, -2, 0.4, -0.4],
       [1, -1, 2.2, -2.2]]

% Singular values: 5, 4, 3
U*diag([5, 4, 3])*V'
      [[2.5, 2.5, 2.5, 2.5],
       [2.4, -2.4, 0, 0],
       [0.7, -0.7, 2.5, -2.5]]

% Singular values: 5, 5, 1
U*diag([5, 5, 1])*V'
      [[2.5, 2.5, 2.5, 2.5],
       [1.9, -1.9, 1.1, -1.1],
       [1.7, -1.7, 2.3, -2.3]]

% Singular values: 5, 5, 2
U*diag([5, 5, 2])*V'
      [[2.5, 2.5, 2.5, 2.5],
       [2.3, -2.3, 0.7, -0.7],
       [1.4, -1.4, 2.6, -2.6]]

% Singular values: 5, 5, 3
U*diag([5, 5, 3])*V'
      [[2.5, 2.5, 2.5, 2.5],
       [2.7, -2.7, 0.3, -0.3],
       [1.1, -1.1, 2.9, -2.9]]

% Singular values: 5, 5, 4
U*diag([5, 5, 4])*V'
      [[2.5, 2.5, 2.5, 2.5],
       [3.1, -3.1, -0.1, 0.1],
       [0.8, -0.8, 3.2, -3.2]]