EVD to SVD

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EVD(EigenValue Decomposition): Symmetric real n by n matrix A = VDV’. Here columns of V are eigenvectors for A and form an orthnormal basis. D is diagonal matrix with entries being eigenvalues of A.

SVD(Sigular Value Decomposition): Arbitrary real m by n matrix A = UDV’. There are orthogonal matices U(m by m) and V(n by n). Columns of U are called left singular vectors while columns of V are called right singular vectors. D is matrix with diagonal entries being singular values of A.

By thinking of matrices as linear tranformations, EVD breaks the transformation of A into n transformations. Here each of them either dilates or contracts one dimension according to the magnitudes of the eigenvalues. Moreover, the basis is orthonormal.  举例来说，对于二维向量v，A把v变换为w，维度并没有变化，w还是在平面上。

Analogously, SVD breaks the transformation of A into

EVD to SVD

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原文地址：https://www.cnblogs.com/prototyping/p/10332207.html