Low performance for Hermitian-times-adjoint matrix multiply

[mikmoore]

From this Discourse thread:

julia> using BenchmarkTools

julia> X = randn(100,100); Y = randn(100,100);

julia> @btime *($X, $Y);
  17.094 μs (3 allocations: 78.21 KiB)

julia> @btime *($X, $Y');
  17.559 μs (3 allocations: 78.21 KiB)

julia> @btime *(Hermitian($X), $Y);
  14.791 μs (3 allocations: 78.21 KiB)

julia> @btime *(Hermitian($X), $Y'); # <-------- oof
  406.171 μs (3 allocations: 78.21 KiB)

julia> @btime *(Matrix(Hermitian($X)), $Y'); # if nothing else we should at least do this?
  22.360 μs (6 allocations: 156.42 KiB)

The hermitian-times-adjoint multiply is 25x slower than hermitian-times-matrix at this size and the gap appears to grow when larger. My guess is a BLAS method was missed and so it hits a fallback. There might be other similar methods that are mis-routed as well.

[jishnub]

This is addressed on v1.12, using exactly the solution proposed above:

julia> @btime *(Hermitian($X), $Y');
  699.341 μs (3 allocations: 78.21 KiB) # v1.11
  34.671 μs (6 allocations: 156.42 KiB) # v1.12