Add methods for matrix multiplication involving Hessenberg orthogonality matrix

[RJDennis]

Some matrix multiplications involving the Hessenberg orthogonality matrix are not currently possible, i.e., the last line in the following code-segment errors:

g =complex(randn(2,2))
h = randn(2,2)
(v,s) = hessenberg(h)
k = v*g

With g a complex matrix and v the Hessenberg orthogonality matrix, the proposal is to add methods to allow:

v*g
v'g
g*v
g*v'

[stevengj]

As mentioned in #38441 (comment), this involves supplementing the LAPACK calls here:

julia/stdlib/LinearAlgebra/src/hessenberg.jl

Lines 488 to 509 in 3365b4e

	lmul!(Q::BlasHessenbergQ{T,false}, X::StridedVecOrMat{T}) where {T<:BlasFloat} =
	LAPACK.ormhr!('L', 'N', 1, size(Q.factors, 1), Q.factors, Q.τ, X)
	rmul!(X::StridedMatrix{T}, Q::BlasHessenbergQ{T,false}) where {T<:BlasFloat} =
	LAPACK.ormhr!('R', 'N', 1, size(Q.factors, 1), Q.factors, Q.τ, X)
	lmul!(adjQ::Adjoint{<:Any,<:BlasHessenbergQ{T,false}}, X::StridedVecOrMat{T}) where {T<:BlasFloat} =
	(Q = adjQ.parent; LAPACK.ormhr!('L', ifelse(T<:Real, 'T', 'C'), 1, size(Q.factors, 1), Q.factors, Q.τ, X))
	rmul!(X::StridedMatrix{T}, adjQ::Adjoint{<:Any,<:BlasHessenbergQ{T,false}}) where {T<:BlasFloat} =
	(Q = adjQ.parent; LAPACK.ormhr!('R', ifelse(T<:Real, 'T', 'C'), 1, size(Q.factors, 1), Q.factors, Q.τ, X))
	lmul!(Q::BlasHessenbergQ{T,true}, X::StridedVecOrMat{T}) where {T<:BlasFloat} =
	LAPACK.ormtr!('L', Q.uplo, 'N', Q.factors, Q.τ, X)
	rmul!(X::StridedMatrix{T}, Q::BlasHessenbergQ{T,true}) where {T<:BlasFloat} =
	LAPACK.ormtr!('R', Q.uplo, 'N', Q.factors, Q.τ, X)
	lmul!(adjQ::Adjoint{<:Any,<:BlasHessenbergQ{T,true}}, X::StridedVecOrMat{T}) where {T<:BlasFloat} =
	(Q = adjQ.parent; LAPACK.ormtr!('L', Q.uplo, ifelse(T<:Real, 'T', 'C'), Q.factors, Q.τ, X))
	rmul!(X::StridedMatrix{T}, adjQ::Adjoint{<:Any,<:BlasHessenbergQ{T,true}}) where {T<:BlasFloat} =
	(Q = adjQ.parent; LAPACK.ormtr!('R', Q.uplo, ifelse(T<:Real, 'T', 'C'), Q.factors, Q.τ, X))
	lmul!(Q::HessenbergQ{T}, X::Adjoint{T,<:StridedVecOrMat{T}}) where {T} = rmul!(X', Q')'
	rmul!(X::Adjoint{T,<:StridedMatrix{T}}, Q::HessenbergQ{T}) where {T} = lmul!(Q', X')'
	lmul!(adjQ::Adjoint{<:Any,<:HessenbergQ{T}}, X::Adjoint{T,<:StridedVecOrMat{T}}) where {T} = rmul!(X', adjQ')'
	rmul!(X::Adjoint{T,<:StridedMatrix{T}}, adjQ::Adjoint{<:Any,<:HessenbergQ{T}}) where {T} = lmul!(adjQ', X')'

with fallback replacements written in pure Julia. i.e. re-implementing the ormhr function.

We may already have most of the the necessary pieces in our QR code?