julia> using TensorKit, MatrixAlgebraKit

julia> a = randn(ℂ^3 → (ℂ^3)')
3←3 TensorMap{Float64, ComplexSpace, 1, 1, Vector{Float64}}:
 codomain: ⊗((ℂ^3)')
 domain: ⊗(ℂ^3)
 blocks: 
 * Trivial() => 3×3 reshape(view(::Vector{Float64}, 1:9), 3, 3) with eltype Float64:
 -1.1574    -0.507056   -1.3797
 -1.30112    0.316528   -0.687373
 -0.619306  -0.0116334   0.515426

julia> a2 = project_hermitian(a)
3←3 TensorMap{Float64, ComplexSpace, 1, 1, Vector{Float64}}:
 codomain: ⊗((ℂ^3)')
 domain: ⊗(ℂ^3)
 blocks: 
 * Trivial() => 3×3 reshape(view(::Vector{Float64}, 1:9), 3, 3) with eltype Float64:
 -1.1574    -0.904089  -0.999501
 -0.904089   0.316528  -0.349503
 -0.999501  -0.349503   0.515426

julia> ishermitian(a2)
false