MAT223H1 Chapter Notes - Chapter 8.4: Diagonal Matrix, Symmetric Matrix, Asteroid Family
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In this section we work on a very important factorization which is a generalization of the diagonalization procedure. This new approach is called the singular value decompo- sition (svd). The singular value decomposition is a factorization pro- cess that can be applied to any matrix, even those that are not square. Let a be a n m matrix. If n m, then the singular value decomposition (svd) is the factorization of a as the product. A = u v t where: u is a n n orthogonal matrix, is an n m matrix of the form (cid:20) 0(n m)m where d is a diagonal matrix with. M (1) (2) (3) (cid:21) (cid:3) and 1 2 . M 0 are the singular values of a. The singular values are related to the eigenvalues of the matrix at a by the expression (cid:112) I: v is a m m orthogonal matrix.
