vectorsandmore.md

Preliminary notes:

• Abstract is left out of the type names for conciseness
• We need a Covector (or alternative name) type for theoretical reasons, i.e. to make dispatch able to distinguish between all of the following operations and to have methods with a different return type for these different operations.
• We need a LazyTranspose type (and Covector) for computational efficiency, i.e. to link with BLAS, to replace the current Ac_mul_Bt etc family.
• Alternatives with ? ... ? are not very likely, just to point out the possibility.

Given vectors v and w, linear maps with matrix representation A and B, and a covector z

abstract operation	julia ascii	julia method	possible alternative or unicode equivalent
Inner product	dot(v, w)	dot(::Vector, ::Vector) -> Number	v ∙ w
Mapping vector to covector	z = v'	transpose(::Vector) -> Covector	Covector(v), ? dot(v) (incomplete inner product) ?
Mapping covector to vector	v = z'	transpose(::Covector) -> Vector
Applying a covector	z * w	*(::Covector, ::Vector) -> Number	? z(w) ?
Combined operation	v'w = v' * w = dot(v,w)	no special method required
Applying linear map	A * v	*(::Matrix, ::Vector) -> Vector	? A(v) ?
Transpose of a map	B = A.'	transpose(::Matrix) -> LazyTranpose
Adjoint of a map	B = A'	ctranspose(::Matrix) -> LazyTranpose
Composing linear maps	A * B	*(::Matrix, ::Matrix) -> Matrix	? A ◦ B ?
Composing linear form and linear map	z * A	*(::Covector,::Matrix) -> Covector	? z ◦ A ?
Combined operation	A.' * v	no special method required
Combined operation	A' * v	no special method required
Combined operation	v'A = v' * A	no special method required	? dot(v,A) = v ∙ A ?
Tensor product between vectors	kron(v, w)	kron(::Vector, ::Vector) -> Vector	v ⊗ w ?
Outer product = tensor product between vector and covector	w * z = w * v'	*(::Vector, ::Covector) -> Matrix	w ⊗ z ?