Statement of The Theorem
Suppose M is an m×n matrix whose entries come from the field K, which is either the field of real numbers or the field of complex numbers. Then there exists a factorization of the form
where U is an m×m unitary matrix over K, the matrix Σ is an m×n diagonal matrix with nonnegative real numbers on the diagonal, and the n×n unitary matrix V* denotes the conjugate transpose of the n×n unitary matrix V. Such a factorization is called the singular value decomposition of M.
The diagonal entries of Σ are known as the singular values of M. A common convention is to list the singular values in descending order. In this case, the diagonal matrix Σ is uniquely determined by M (though the matrices U and V are not).
Read more about this topic: Singular Value Decomposition
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