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Indeed, this is a corner case but there is no reason for it to error (it works in GenericLinearAlgebra, i.e. for non-BLAS eltypes).
julia> ones(2, 0)*LinearAlgebra.hessenberg(zeros(0,0)).Q
ERROR: DimensionMismatch: tau has length 0, needs -1
Stacktrace:
[1] ormhr!(side::Char, trans::Char, ilo::Int64, ihi::Int64, A::Matrix{Float64}, tau::Vector{Float64}, C::Matrix{Float64})
@ LinearAlgebra.LAPACK ~/.julia/juliaup/julia-1.11.4+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/lapack.jl:6131
[2] mul!(C::Matrix{Float64}, A::Matrix{Float64}, Q::LinearAlgebra.HessenbergQ{Float64, Matrix{…}, Vector{…}, false})
@ LinearAlgebra ~/.julia/juliaup/julia-1.11.4+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/abstractq.jl:0
[3] *(A::Matrix{Float64}, Q::LinearAlgebra.HessenbergQ{Float64, Matrix{Float64}, Vector{Float64}, false})
@ LinearAlgebra ~/.julia/juliaup/julia-1.11.4+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/abstractq.jl:204
[4] top-level scope
@ REPL[228]:1
Some type information was truncated. Use `show(err)` to see complete types.Metadata
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