Intuitively, what is the difference between Eigendecomposition and Singular Value Decomposition?

Abstraction: Mathematical relationship showing SVD generalizes eigendecomposition to rectangular matrices

Key points:

• Eigendecomposition (symmetric case): A = O diag(lambda_1,...,lambda_n) O' where O is orthogonal; classifies symmetric matrices under "orthogonal similar" equivalence
• SVD stems directly from eigendecomposition: singular values of B are square roots of eigenvalues of B'B (which is symmetric positive semi-definite)
• SVD: B = O1 diag(mu_1,...,mu_r, 0) O2 for any rectangular B in R^(m x n); classifies all matrices under "orthogonal equivalent" equivalence
• Key distinction: eigendecomposition uses a single orthogonal matrix (same basis change on both sides); SVD uses two different orthogonal matrices (one per mode)
• Both seek a "representative element" (diagonal matrix) under their respective equivalence relations; eigenvalues and singular values are the respective maximal invariants
• SVD is strictly more general: every real matrix has an SVD, but only square matrices have eigendecompositions, and only symmetric ones are guaranteed real diagonal form

Connections: Eigendecomposition · Singular Value Decomposition · Linear Algebra

Source: http://math.stackexchange.com/questions/320220/intuitively-what-is-the-difference-between-eigendecomposition-and-singular-valu