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

V =
      [[0.1, 0.5, 0.5, 0.7], [0.5, -0.1, -0.7, 0.5], [0.5, 0.7, -0.1, -0.5], [0.7, -0.5, 0.5, -0.1]]

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

% Singular values: 2, 2, 1
U*diag([2, 2, 1])*V'
      [[0.2, 1, 1, 1.4], [1, -0.68, 0.76, -0.2], [0.5, 0.26, 1.18, -1.1]]

% Singular values: 3, 2, 1
U*diag([3, 2, 1])*V'
      [[0.3, 1.5, 1.5, 2.1], [1, -0.68, 0.76, -0.2], [0.5, 0.26, 1.18, -1.1]]

% Singular values: 4, 2, 1
U*diag([4, 2, 1])*V'
      [[0.4, 2, 2, 2.8], [1, -0.68, 0.76, -0.2], [0.5, 0.26, 1.18, -1.1]]

% Singular values: 4, 3, 1
U*diag([4, 3, 1])*V'
      [[0.4, 2, 2, 2.8], [1.3, -0.74, 1.18, -0.5], [0.9, 0.18, 1.74, -1.5]]

% Singular values: 4, 3, 2
U*diag([4, 3, 2])*V'
      [[0.4, 2, 2, 2.8], [1.7, -1.3, 1.1, -0.1], [0.6, 0.6, 1.8, -1.8]]

% Singular values: 4, 4, 1
U*diag([4, 4, 1])*V'
      [[0.4, 2, 2, 2.8], [1.6, -0.8, 1.6, -0.8], [1.3, 0.1, 2.3, -1.9]]

% Singular values: 4, 4, 2
U*diag([4, 4, 2])*V'
      [[0.4, 2, 2, 2.8], [2, -1.36, 1.52, -0.4], [1, 0.52, 2.36, -2.2]]

% Singular values: 5, 2, 1
U*diag([5, 2, 1])*V'
      [[0.5, 2.5, 2.5, 3.5], [1, -0.68, 0.76, -0.2], [0.5, 0.26, 1.18, -1.1]]

% Singular values: 5, 3, 1
U*diag([5, 3, 1])*V'
      [[0.5, 2.5, 2.5, 3.5], [1.3, -0.74, 1.18, -0.5], [0.9, 0.18, 1.74, -1.5]]

% Singular values: 5, 3, 2
U*diag([5, 3, 2])*V'
      [[0.5, 2.5, 2.5, 3.5], [1.7, -1.3, 1.1, -0.1], [0.6, 0.6, 1.8, -1.8]]

% Singular values: 5, 4, 1
U*diag([5, 4, 1])*V'
      [[0.5, 2.5, 2.5, 3.5], [1.6, -0.8, 1.6, -0.8], [1.3, 0.1, 2.3, -1.9]]

% Singular values: 5, 4, 2
U*diag([5, 4, 2])*V'
      [[0.5, 2.5, 2.5, 3.5], [2, -1.36, 1.52, -0.4], [1, 0.52, 2.36, -2.2]]

% Singular values: 5, 4, 3
U*diag([5, 4, 3])*V'
      [[0.5, 2.5, 2.5, 3.5], [2.4, -1.92, 1.44, 0], [0.7, 0.94, 2.42, -2.5]]