If a matrix has a matrix of eigenvectors that is not invertible (for example, the matrix has the noninvertible system of eigenvectors ), then does not have an eigen decomposition. However, if is an real matrix with , then can be written using a so-called singular value decomposition of the form
(1)
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Note that there are several conflicting notational conventions in use in the literature. Press et al. (1992) define to be an matrix, as , and as . However, the Wolfram Language defines as an , as , and as . In both systems, and have orthogonal columns so that
(2)
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and
(3)
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(where the two identity matrices may have different dimensions), and has entries only along the diagonal.
For a complex matrix , the singular value decomposition is a decomposition into the form
(4)
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where and are unitary matrices, is the conjugate transpose of , and is a diagonal matrix whose elements are the singular values of the original matrix. If is a complex matrix, then there always exists such a decomposition with positive singular values (Golub and Van Loan 1996, pp. 70 and 73).
Singular value decomposition is implemented in the Wolfram Language as SingularValueDecomposition[m], which returns a list U, D, V, where U and V are matrices and D is a diagonal matrix made up of the singular values of .