Notes About Singular Value Decomposition

A brief summary of SVD:

An original matrix Amn is represented as a muliplication of three matrices:

Amn = UmmSmnVnnT

The columns of U are the orthonormal engenvectors of AAT descendingly ordered by the corresponding eigenvalues, and the columns of V are the orthonormal engenvectors of ATA descendingly ordered by the corresponding eigenvalues. This also suggests that U and V are orthonormal (orthogonal) matrices. A characteristic of the two matrices is that the non-zero eigenvalues of U and V are always the same. S is a diagonal matrix containing the square roots of eigenvalues from U and V in descending order. The diagonal entries in S are the singular values of A, the columns in U are called left singular vectors, and the columns in V are called right singular vectors.

时间： 2025-01-18 09:56:01

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