.. _svdVectorChange:

How SVD Changes Vectors
=======================

With the exception of rotation matrices and eigenvectors, we normally
see both rotation and stretching when a point or vector is multiplied by
a matrix. When the matrix is represented by its SVD factors, we see
rotation, stretching, and again rotation when we multiply sequentially
by :math:`\mathbf{V}^T`, :math:`\mathbf{\Sigma}`, and
:math:`\mathbf{U}`. 
The ``showSVD`` function may be used to experiment with several
:math:`2{\times}2` matrices to see the rotation, stretching, and rotation. The
demonstration starts with a set of points on a circle. In the end, the circle
pattern will be changed to show a rotated ellipse. Figures
:numref:`fig-SVDrotV`, and :numref:`fig-SVDrotSigU` show example plots. The
plots will be different for each matrix supplied.

::

   function show_SVD(A)
   % SHOW_SVD - A demonstration of how the U, S, and V matrices
   %   from SVD rotate, stretch, and rotate vectors making a circle.
   %   Try several 2x2 matrices to see how each behaves.

   theta = linspace(0, 2*pi, 30);
   x = [cos(theta); sin(theta)];

   [U,S,V] = svd(A);

   Vx = V'*x;
   figure, plot(x(1,:), x(2,:), '*')
   hold on
   scatter(Vx(1,:), Vx(2,:))
   for z = 1:30
       line([x(1,z), Vx(1,z)], [x(2,z), Vx(2,z)], 'Color', 'k')
   end
   title('Rotation by V^T Matrix')
   hold off

   svx = S*Vx;
   figure, scatter(svx(1,:), svx(2,:), '*')
   daspect([1 1 1])
   usvx = U*svx;
   hold on
   scatter(usvx(1,:), usvx(2,:), 'o')
   title('Stretch by /Sigma and rotation by U')
   hold off

.. _fig-SVDrotV:

.. figure:: SVDrotV.png 
   :align: center
   :width: 30%

   When each point is multiplied by :math:`\mathbf{V}^T` the points
   rotate around the circle. The lines show the initial and final positions
   of the points. 

.. _fig-SVDrotSigU:

.. figure:: SVDrotSigU.png
   :align: center
   :width: 30%

   When each point is multiplied by :math:`\mathbf{\Sigma}`, the points around
   the circle are moved out from the center to form an ellipse. Then the
   ellipse is rotated when the points are multiplied by :math:`\bf{U}`.