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A Constructive Approach to Singular Value Decomposition and Symmetric Schur Factorization

Geometrical proof of the SVD that avoids use of the symmetric Schur factorization and instead obtains that as a corollary. This approach emphasizes the fact that the image of a sphere under a linear transformation is an ellipsoid.

This is an article from The American Mathematical Monthly, Vol. 112, No. 4 (Apr., 2005), pp. 358-363.

Identifier: 
http://www.jstor.org/stable/30037473
Rating: 
Average: 3.3 (3 votes)
Creator(s): 
John Clifford, David James, Michael Lachance and Joan Remski
Cataloger: 
Daniel Drucker
Publisher: 
MAA, Amer Math Monthly, Vol. 112, No. 4 (Apr. 2005), 358–363.
Rights: 
John Clifford, David James, Michael Lachance, & Joan Remski

Comments

Anonymous's picture

This is an article from The American Mathematical Monthly, Vol. 112, No. 4 (Apr., 2005), pp. 358-363. The authors prove the existance of the singular value decomposition without reference to eigenvalues and eigenvectors, by instead developing a constructive proof. They then show that a small modification of their argument can be used to prove the existence of the symmetric Schur factorization. The article is clearly written and is quite interesting. However, the level is such that most students in a typical linear algebra course would need considerable guidance to get through it. It seems like it would be well suited for a more advanced undergraduate with a deeper mathematics background.

drucker@math.wayne.edu's picture

Not suitable for first course in linear algebra because the proof assumes some basic analysis on Euclidean spaces, such as the fact that a continuous function on a closed bounded set attains a maximum value. Probably fine for a second course on linear algebra.

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