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

V =
      [[0.1, 0.3, 0.3, 0.9], [0.3, -0.1, -0.9, 0.3], [0.3, 0.9, -0.1, -0.3], [0.9, -0.3, 0.3, -0.1]]

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

% Singular values: 2, 2, 1
U*diag([2, 2, 1])*V'
      [[0.2, 0.6, 0.6, 1.8], [0.6, -0.84, 1, -0.12], [0.3, 0.38, 1.5, -0.66]]

% Singular values: 3, 2, 1
U*diag([3, 2, 1])*V'
      [[0.3, 0.9, 0.9, 2.7], [0.6, -0.84, 1, -0.12], [0.3, 0.38, 1.5, -0.66]]

% Singular values: 3, 3, 2
U*diag([3, 3, 2])*V'
      [[0.3, 0.9, 0.9, 2.7], [1.02, -1.62, 1.46, -0.06], [0.36, 0.84, 2.28, -1.08]]

% Singular values: 4, 2, 1
U*diag([4, 2, 1])*V'
      [[0.4, 1.2, 1.2, 3.6], [0.6, -0.84, 1, -0.12], [0.3, 0.38, 1.5, -0.66]]

% Singular values: 4, 3, 1
U*diag([4, 3, 1])*V'
      [[0.4, 1.2, 1.2, 3.6], [0.78, -0.9, 1.54, -0.3], [0.54, 0.3, 2.22, -0.9]]

% Singular values: 4, 3, 2
U*diag([4, 3, 2])*V'
      [[0.4, 1.2, 1.2, 3.6], [1.02, -1.62, 1.46, -0.06], [0.36, 0.84, 2.28, -1.08]]

% Singular values: 4, 4, 1
U*diag([4, 4, 1])*V'
      [[0.4, 1.2, 1.2, 3.6], [0.96, -0.96, 2.08, -0.48], [0.78, 0.22, 2.94, -1.14]]

% Singular values: 4, 4, 2
U*diag([4, 4, 2])*V'
      [[0.4, 1.2, 1.2, 3.6], [1.2, -1.68, 2, -0.24], [0.6, 0.76, 3, -1.32]]

% Singular values: 4, 4, 3
U*diag([4, 4, 3])*V'
      [[0.4, 1.2, 1.2, 3.6], [1.44, -2.4, 1.92, 0], [0.42, 1.3, 3.06, -1.5]]

% Singular values: 5, 2, 1
U*diag([5, 2, 1])*V'
      [[0.5, 1.5, 1.5, 4.5], [0.6, -0.84, 1, -0.12], [0.3, 0.38, 1.5, -0.66]]

% Singular values: 5, 3, 1
U*diag([5, 3, 1])*V'
      [[0.5, 1.5, 1.5, 4.5], [0.78, -0.9, 1.54, -0.3], [0.54, 0.3, 2.22, -0.9]]

% Singular values: 5, 3, 2
U*diag([5, 3, 2])*V'
      [[0.5, 1.5, 1.5, 4.5], [1.02, -1.62, 1.46, -0.06], [0.36, 0.84, 2.28, -1.08]]

% Singular values: 5, 4, 1
U*diag([5, 4, 1])*V'
      [[0.5, 1.5, 1.5, 4.5], [0.96, -0.96, 2.08, -0.48], [0.78, 0.22, 2.94, -1.14]]

% Singular values: 5, 4, 2
U*diag([5, 4, 2])*V'
      [[0.5, 1.5, 1.5, 4.5], [1.2, -1.68, 2, -0.24], [0.6, 0.76, 3, -1.32]]

% Singular values: 5, 4, 3
U*diag([5, 4, 3])*V'
      [[0.5, 1.5, 1.5, 4.5], [1.44, -2.4, 1.92, 0], [0.42, 1.3, 3.06, -1.5]]

% Singular values: 5, 5, 4
U*diag([5, 5, 4])*V'
      [[0.5, 1.5, 1.5, 4.5], [1.86, -3.18, 2.38, 0.06], [0.48, 1.76, 3.84, -1.92]]