Another means of examining singular value decompositions, described by Hern and Long (1991), is illustrated through the **Hern & Long SVD** tool, in which the decomposition is examined from the perspective of matrices of transformations that rotate, reflect, and stretch a unit circle to yield an ellipse. Rather than being an exploratory tool, this is an explanatory tool helpful for classroom discussions of the components of the SVD.

Consistent with the description of Hern and Long (1991), this tool contains a several regions that depict different aspects of the geometry associated with the singular value decomposition. As with most of our tools, the yellow box controls the column vectors defining the matrix of transformation. In particular, the **green** vector is the first column, and the **blue** vector is the second column. By grabbing the ends of these two vectors, you can construct any 2x2 matrix.

Next to the yellow box is the domain of the transformation that contains four vectors, **x**, **p**, **v1**, and **v2**. Vectors **x** and **p** are movable, with **p** restricted to the unit circle. Vectors **v1** and **v2** are restricted to the unit circle and dependent on the matrix of transformation. Specifically, **v1** and **v2** are the eigenvectors of the symmetric matrix *A*^{T}*A*. The transformation defined by the matrix *V*^{T} rotates the vectors **v1** and **v2** to the base vectors **e**_{1} and **e**_{2}. Then, the matrix *S* stretches these base vectors by the factors that are the lengths of the axes (and the positive square roots of the nonzero eigenvalues of *AA*^{T}). Finally, the matrix *U* rotates these to *A***v1** and *A***v2**. Consequently, the matrix *A* can be written as *V*^{T}*SU,* where the entries *s*_{i} of *S* are called the singular values of *A,* and *A* = *V*^{T}*SU* is called the singular value decomposition of *A*.

Open Hern & Long SVD in new window

Note: The "?" at the bottom right-hand corner of the workspace is a link to Key Curriculum Press and its About *JavaSketchpad* web page.

Next or or page: **19.** Conclusion