Applications of singular-value decomposition (SVD)

Autor: Gennadi Malaschonok, Alkiviadis G. Akritas
Rok vydání: 2004
Předmět:
Zdroj: Mathematics and Computers in Simulation. 67:15-31
ISSN: 0378-4754
DOI: 10.1016/j.matcom.2004.05.005
Popis: Let A be an m×n matrix with m≥n. Then one form of the singular-value decomposition of A is A=U T ΣV, where U and V are orthogonal and Σ is square diagonal. That is, UUT=Irank(A), VVT=Irank(A), U is rank(A)×m, V is rank(A)×n and Σ= σ 1 0 ⋯ 0 0 0 σ 2 ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ σ rank (A)−1 0 0 0 ⋯ 0 σ rank (A) is a rank(A)×rank(A) diagonal matrix. In addition σ1≥σ2≥⋯≥σrank(A)>0. The σi’s are called the singular values of A and their number is equal to the rank of A. The ratio σ1/σrank(A) can be regarded as a condition number of the matrix A. It is easily verified that the singular-value decomposition can be also written as A=U T ΣV=∑ i=1 rank (A) σ i u i T v i . The matrix uiTvi is the outer product of the i-th row of U with the corresponding row of V. Note that each of these matrices can be stored using only m+n locations rather than mn locations. Using both forms presented above—and following Jerry Uhl’s beautiful approach in the Calculus and Mathematica book series [Matrices, Geometry & Mathematica, Math Everywhere Inc., 1999]—we show how SVD can be used as a tool for teaching Linear Algebra geometrically, and then apply it in solving least-squares problems and in data compression. In this paper we used the Computer Algebra system Mathematica to present a purely numerical problem. In general, the use of Computer Algebra systems has greatly influenced the teaching of mathematics, allowing students to concentrate on the main ideas and to visualize them.
Databáze: OpenAIRE
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