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 integer coefficients.
