Singular Value

There are two types of singular values, one in the context of elliptic integrals, and the other in linear algebra.

For a square matrix , the square roots of the eigenvalues of , where  is the conjugate transpose, are called singular values (Marcus and Minc 1992, p. 69). The so-called singular value decomposition of a complex matrix  is given by

(1)

where  and  are unitary matrices and  is a diagonal matrix whose elements are the singular values of  (Golub and Van Loan 1996, pp. 70 and 73). Singular values are returned by the command SingularValueList[m].

If

(2)

where  is a unitary matrix and  is a Hermitian matrix, then the eigenvalues of  are the singular values of .

For elliptic integrals, a elliptic modulus  such that

(3)

where  is a complete elliptic integral of the first kind, and . The elliptic lambda function  gives the value of . Abel (quoted in Whittaker and Watson 1990, p. 525) proved that if  is an integer, or more generally whenever

(4)

where , , , , and  are integers, then the elliptic modulus  is the root of an algebraic equation with integercoefficients.

REFERENCES:

Golub, G. H. and Van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins, 1996.

Marcus, M. and Minc, H. Introduction to Linear Algebra. New York: Dover, p. 191, 1988.

Marcus, M. and Minc, H. A Survey of Matrix Theory and Matrix Inequalities. New York: Dover, p. 69, 1992.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 524-528, 1990.