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Journal of Online Mathematics and its Applications
Tool Building: Web-based Linear Algebra Modules
Hern & Long SVD Tool
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 ATA. The transformation defined by the matrix VT rotates the vectors v1 and v2 to the base vectors e1 and e2. 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 AAT). Finally, the matrix U rotates these to Av1 and Av2. Consequently, the matrix A can be written as VTSU, where the entries si of S are called the singular values of A, and A = VTSU is called the singular value decomposition of A.
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Note: The "?" at the bottom right-hand corner of the workspace is a link to Key Curriculum Press and its About JavaSketchpad web page.
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