Determinant
.

Determinant (.
) is a primitive dyadic operator that takes two dyadic functions for operands and produces a monadic function. Much like Inner Product for matrix products, it generalizes the matrix determinant, so that .×
computes the determinant and +.×
the permanent. Determinant has been implemented in SHARP APL, NARS2000, and J; however, the J definition differs by factoring out a reduction from the definition, so that / .*
is the matrix determinant.
Unlike the inner product operator, which performs arithmetic operations on square matrices of side length given arithmetic operands, definitions of the determinant use a superpolynomial number of evaluations, for example in SHARP APL's definition. Computing the permanent in particular is not believed to be possible in polynomial time, which would rule out a polynomialtime definition for any generalization of it. The specific determinant function .×
can be computed much faster (for example by LU decomposition in ), and is always handled specially for performance. Other combinations of operands may also have optimized implementations.
The determinant operator was described by Ken Iverson in "DeterminantLike Functions Produced by the Dot Operator"^{[1]} and also implemented in SHARP APL^{[2]} in 1982. It also appears in Iverson's later publications Rationalized APL and A Dictionary of APL.
Documentation
References
 ↑ Ken Iverson. SATN42: DeterminantLike Functions Produced by the Dot Operator. 19820401.
 ↑ IPSA Newsletter July/August 1982 (pdf)