Pattern Recognition, 4th Edition

6.4. The Singular Value Decomposition

The singular value decomposition of a matrix is one of the most elegant and powerful algorithms in linear algebra, and it has been extensively used for rank and dimension reduction in pattern recognition and information retrieval applications. Given a l × n matrix X of rank r (obviously r ≤ min{l, n}), we will show that there exist unitary matrices U and V of dimensions l × l and n × n, respectively, so that(6.24)where Λ½ is the r × r diagonal matrix with elements , and λⁱ are the r nonzero eigenvalues of the ...