define dot between AbstractMatrix and UniformScaling

NEWS.md

@@ -74,6 +74,7 @@ Standard library changes
 * OpenBLAS is updated to 0.3.13. ([#39216])
 * SuiteSparse is updated to 5.8.1. ([#39455])
 * `cis(A)` now supports matrix arguments ([#40194]).
+* `dot` now supports `UniformScaling` with `AbstractMatrix` ([#40250]).
 
 #### Markdown
 

stdlib/LinearAlgebra/src/uniformscaling.jl

@@ -480,6 +480,9 @@ Array(s::UniformScaling, dims::Dims{2}) = Matrix(s, dims)
 Diagonal{T}(s::UniformScaling, m::Integer) where {T} = Diagonal{T}(fill(T(s.λ), m))
 Diagonal(s::UniformScaling, m::Integer) = Diagonal{eltype(s)}(s, m)
 
+dot(A::AbstractMatrix, J::UniformScaling) = dot(tr(A), J.λ)
+dot(J::UniformScaling, A::AbstractMatrix) = dot(J.λ, tr(A))
+
 dot(x::AbstractVector, J::UniformScaling, y::AbstractVector) = dot(x, J.λ, y)
 dot(x::AbstractVector, a::Number, y::AbstractVector) = sum(t -> dot(t[1], a, t[2]), zip(x, y))
 dot(x::AbstractVector, a::Union{Real,Complex}, y::AbstractVector) = a*dot(x, y)

stdlib/LinearAlgebra/test/uniformscaling.jl

@@ -462,6 +462,20 @@ end
     @test I(3) == [1 0 0; 0 1 0; 0 0 1]
 end
 
+@testset "dot" begin
+    A = randn(3, 3)
+    λ = randn()
+    J = UniformScaling(λ)
+    @test dot(A, J) ≈ dot(J, A)
+    @test dot(A, J) ≈ tr(A' * J)
+
+    A = rand(ComplexF64, 3, 3)
+    λ = randn() + im * randn()
+    J = UniformScaling(λ)
+    @test dot(A, J) ≈ conj(dot(J, A))
+    @test dot(A, J) ≈ tr(A' * J)
+end
+
 @testset "generalized dot" begin
     x = rand(-10:10, 3)
     y = rand(-10:10, 3)