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.
