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

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

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

% Singular values: 2, 2, 1
U*diag([2, 2, 1])*V'
      [[1, 1, 1, 1], [0.92, -0.92, -0.44, 0.44], [1.06, -1.06, 0.58, -0.58]]

% Singular values: 3, 2, 1
U*diag([3, 2, 1])*V'
      [[1.5, 1.5, 1.5, 1.5], [0.92, -0.92, -0.44, 0.44], [1.06, -1.06, 0.58, -0.58]]

% Singular values: 3, 3, 2
U*diag([3, 3, 2])*V'
      [[1.5, 1.5, 1.5, 1.5], [1.42, -1.42, -0.94, 0.94], [1.56, -1.56, 1.08, -1.08]]

% Singular values: 4, 2, 1
U*diag([4, 2, 1])*V'
      [[2, 2, 2, 2], [0.92, -0.92, -0.44, 0.44], [1.06, -1.06, 0.58, -0.58]]

% Singular values: 4, 3, 1
U*diag([4, 3, 1])*V'
      [[2, 2, 2, 2], [1.34, -1.34, -0.38, 0.38], [1.62, -1.62, 0.66, -0.66]]

% Singular values: 4, 3, 2
U*diag([4, 3, 2])*V'
      [[2, 2, 2, 2], [1.42, -1.42, -0.94, 0.94], [1.56, -1.56, 1.08, -1.08]]

% Singular values: 4, 4, 2
U*diag([4, 4, 2])*V'
      [[2, 2, 2, 2], [1.84, -1.84, -0.88, 0.88], [2.12, -2.12, 1.16, -1.16]]

% Singular values: 4, 4, 3
U*diag([4, 4, 3])*V'
      [[2, 2, 2, 2], [1.92, -1.92, -1.44, 1.44], [2.06, -2.06, 1.58, -1.58]]

% Singular values: 5, 2, 1
U*diag([5, 2, 1])*V'
      [[2.5, 2.5, 2.5, 2.5], [0.92, -0.92, -0.44, 0.44], [1.06, -1.06, 0.58, -0.58]]

% Singular values: 5, 3, 1
U*diag([5, 3, 1])*V'
      [[2.5, 2.5, 2.5, 2.5], [1.34, -1.34, -0.38, 0.38], [1.62, -1.62, 0.66, -0.66]]

% Singular values: 5, 3, 2
U*diag([5, 3, 2])*V'
      [[2.5, 2.5, 2.5, 2.5], [1.42, -1.42, -0.94, 0.94], [1.56, -1.56, 1.08, -1.08]]

% Singular values: 5, 4, 1
U*diag([5, 4, 1])*V'
      [[2.5, 2.5, 2.5, 2.5], [1.76, -1.76, -0.32, 0.32], [2.18, -2.18, 0.74, -0.74]]

% Singular values: 5, 4, 2
U*diag([5, 4, 2])*V'
      [[2.5, 2.5, 2.5, 2.5], [1.84, -1.84, -0.88, 0.88], [2.12, -2.12, 1.16, -1.16]]

% Singular values: 5, 4, 3
U*diag([5, 4, 3])*V'
      [[2.5, 2.5, 2.5, 2.5], [1.92, -1.92, -1.44, 1.44], [2.06, -2.06, 1.58, -1.58]]

% Singular values: 5, 5, 2
U*diag([5, 5, 2])*V'
      [[2.5, 2.5, 2.5, 2.5], [2.26, -2.26, -0.82, 0.82], [2.68, -2.68, 1.24, -1.24]]

% Singular values: 5, 5, 3
U*diag([5, 5, 3])*V'
      [[2.5, 2.5, 2.5, 2.5], [2.34, -2.34, -1.38, 1.38], [2.62, -2.62, 1.66, -1.66]]

% Singular values: 5, 5, 4
U*diag([5, 5, 4])*V'
      [[2.5, 2.5, 2.5, 2.5], [2.42, -2.42, -1.94, 1.94], [2.56, -2.56, 2.08, -2.08]]