fix some issues with equality of factorizations

- `hash` did not respect the type of a factorization, so completely
  different factorizations with the same underlying data would result in
  same `hash` leading to inconsistencies with `isequal`. This likely
  doesn't occur very often in practice, but definitely seems worth
  fixing.
- `==` and `isequal` only returned true if two factorizations are of
  exactly the same type, which is inconsistent with their implementation
  for other objects and with the definition of `hash` for factorizations.
- Equality for `QRCompactWY` did not ignore the subdiagonal entries of
  `T` leading to nondeterministic behavior. Perhaps `T` should be
  directly stored as `UpperTriangular` in `QRCompactWY`, but that seems
  potentially breaking.

Relying on implementation details of `DataType` here is certainly less
than ideal, but I could not come up with a nicer solution.

stdlib/LinearAlgebra/src/factorization.jl

@@ -64,9 +64,17 @@ Factorization{T}(A::Adjoint{<:Any,<:Factorization}) where {T} =
     adjoint(Factorization{T}(parent(A)))
 inv(F::Factorization{T}) where {T} = (n = size(F, 1); ldiv!(F, Matrix{T}(I, n, n)))
 
-Base.hash(F::Factorization, h::UInt) = mapreduce(f -> hash(getfield(F, f)), hash, 1:nfields(F); init=h)
-Base.:(==)(  F::T, G::T) where {T<:Factorization} = all(f -> getfield(F, f) == getfield(G, f), 1:nfields(F))
-Base.isequal(F::T, G::T) where {T<:Factorization} = all(f -> isequal(getfield(F, f), getfield(G, f)), 1:nfields(F))::Bool
+function Base.hash(F::Factorization, h::UInt)
+    return mapreduce(f -> hash(getfield(F, f)), hash, 1:nfields(F); init=hash(typeof(F).name.wrapper, h))
+end
+function Base.:(==)(F::Factorization, G::Factorization)
+    typeof(F).name.wrapper == typeof(G).name.wrapper || return false
+    return all(f -> getfield(F, f) == getfield(G, f), 1:nfields(F))
+end
+function Base.isequal(F::Factorization, G::Factorization)
+    typeof(F).name.wrapper == typeof(G).name.wrapper || return false
+    return all(f -> isequal(getfield(F, f), getfield(G, f)), 1:nfields(F))::Bool
+end
 
 function Base.show(io::IO, x::Adjoint{<:Any,<:Factorization})
     print(io, "Adjoint of ")

stdlib/LinearAlgebra/src/qr.jl

@@ -127,6 +127,16 @@ Base.iterate(S::QRCompactWY) = (S.Q, Val(:R))
 Base.iterate(S::QRCompactWY, ::Val{:R}) = (S.R, Val(:done))
 Base.iterate(S::QRCompactWY, ::Val{:done}) = nothing
 
+function Base.hash(F::QRCompactWY, h::UInt)
+    return hash(F.factors, hash(UpperTriangular(F.T), hash(QRCompactWY, h)))
+end
+function Base.:(==)(A::QRCompactWY, B::QRCompactWY)
+    return A.factors == B.factors && UpperTriangular(A.T) == UpperTriangular(B.T)
+end
+function Base.isequal(A::QRCompactWY, B::QRCompactWY)
+    return isequal(A.factors, B.factors) && isequal(UpperTriangular(A.T), UpperTriangular(B.T))
+end
+
 """
     QRPivoted <: Factorization
 

stdlib/LinearAlgebra/test/factorization.jl

@@ -0,0 +1,50 @@
+# This file is a part of Julia. License is MIT: https://julialang.org/license
+
+module TestFactorization
+using Test, LinearAlgebra
+
+@testset "equality for factorizations - $f" for f in Any[
+    bunchkaufman,
+    cholesky,
+    x -> cholesky(x, Val(true)),
+    eigen,
+    hessenberg,
+    lq,
+    lu,
+    qr,
+    x -> qr(x, ColumnNorm()),
+    svd,
+    schur,
+]
+    A = randn(3, 3)
+    A = A * A' # ensure A is pos. def. and symmetric
+    F, G = f(A), f(A)
+
+    @test F == G
+    @test isequal(F, G)
+    @test hash(F) == hash(G)
+
+    f === hessenberg && continue
+
+    F = typeof(F).name.wrapper(Base.mapany(1:nfields(F)) do i
+        x = getfield(F, i)
+        return x isa AbstractArray{Float64} ? Float32.(x) : x
+    end...)
+    G = typeof(G).name.wrapper(Base.mapany(1:nfields(G)) do i
+        x = getfield(G, i)
+        return x isa AbstractArray{Float64} ? Float64.(Float32.(x)) : x
+    end...)
+
+    @test F == G
+    @test isequal(F, G)
+    @test hash(F) == hash(G)
+end
+
+@testset "hash collisions" begin
+    A, v = randn(2, 2), randn(2)
+    F, G = LQ(A, v), QR(A, v)
+    @test !isequal(F, G)
+    @test hash(F) != hash(G)
+end
+
+end

stdlib/LinearAlgebra/test/testgroups

@@ -25,3 +25,4 @@ givens
 structuredbroadcast
 addmul
 ldlt
+factorization