diff --git a/Project.toml b/Project.toml
index 63743d10..0b0d4c35 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "ImplicitDifferentiation"
 uuid = "57b37032-215b-411a-8a7c-41a003a55207"
 authors = ["Guillaume Dalle", "Mohamed Tarek"]
-version = "0.8.0"
+version = "0.8.1"
 
 [deps]
 ADTypes = "47edcb42-4c32-4615-8424-f2b9edc5f35b"
diff --git a/docs/src/faq.md b/docs/src/faq.md
index 3e7e4edf..bec4b2bf 100644
--- a/docs/src/faq.md
+++ b/docs/src/faq.md
@@ -46,9 +46,6 @@ Say your forward mapping takes multiple inputs and returns multiple outputs, suc
 The trick is to leverage [ComponentArrays.jl](https://github.com/jonniedie/ComponentArrays.jl) to wrap all the inputs inside a single a `ComponentVector`, and do the same for all the outputs.
 See the examples for a demonstration.
 
-!!! warning
-    The default linear operator representation does not support ComponentArrays.jl: you need to select `representation=OperatorRepresentation{:LinearMaps}()` in the [`ImplicitFunction`](@ref) constructor for it to work.
-
 !!! warning
     You may run into issues trying to differentiate through the `ComponentVector` constructor.
     For instance, Zygote.jl will throw `ERROR: Mutating arrays is not supported`.
diff --git a/examples/3_tricks.jl b/examples/3_tricks.jl
index dd43ddd2..a1cf4e89 100644
--- a/examples/3_tricks.jl
+++ b/examples/3_tricks.jl
@@ -42,13 +42,9 @@ function conditions_components(x::ComponentVector, y::ComponentVector, _z)
     return c
 end;
 
-# And build your implicit function like so, switching the operator representation to avoid errors with ComponentArrays.
+# And build your implicit function like so:
 
-implicit_components = ImplicitFunction(
-    forward_components,
-    conditions_components;
-    representation=OperatorRepresentation{:LinearMaps}(),
-);
+implicit_components = ImplicitFunction(forward_components, conditions_components);
 
 # Now we're good to go.
 
diff --git a/src/execution.jl b/src/execution.jl
index 1190bd69..3d5d00e3 100644
--- a/src/execution.jl
+++ b/src/execution.jl
@@ -61,7 +61,7 @@ function build_A_aux(
 end
 
 function build_A_aux(
-    ::OperatorRepresentation{package,symmetric,hermitian},
+    ::OperatorRepresentation{package,symmetric,hermitian,posdef,keep_input_type},
     implicit,
     x,
     y,
@@ -69,7 +69,7 @@ function build_A_aux(
     c,
     args...;
     suggested_backend,
-) where {package,symmetric,hermitian}
+) where {package,symmetric,hermitian,posdef,keep_input_type}
     T = Base.promote_eltype(x, y, c)
     (; conditions, backends, prep_A) = implicit
     actual_backend = isnothing(backends) ? suggested_backend : backends.y
@@ -89,7 +89,13 @@ function build_A_aux(
     prod! = JVP!(f_vec, prep_A_same, actual_backend, y_vec, dy_vec, contexts)
     if package == :LinearOperators
         return LinearOperator(
-            T, length(c), length(y), symmetric, hermitian, prod!; S=typeof(dy_vec)
+            T,
+            length(c),
+            length(y),
+            symmetric,
+            hermitian,
+            prod!;
+            S=keep_input_type ? typeof(dy_vec) : Vector{T},
         )
     elseif package == :LinearMaps
         return FunctionMap{T}(
@@ -99,6 +105,7 @@ function build_A_aux(
             ismutating=true,
             issymmetric=symmetric,
             ishermitian=hermitian,
+            isposdef=posdef,
         )
     end
 end
@@ -136,7 +143,7 @@ function build_Aᵀ_aux(
 end
 
 function build_Aᵀ_aux(
-    ::OperatorRepresentation{package,symmetric,hermitian},
+    ::OperatorRepresentation{package,symmetric,hermitian,posdef,keep_input_type},
     implicit,
     x,
     y,
@@ -144,7 +151,7 @@ function build_Aᵀ_aux(
     c,
     args...;
     suggested_backend,
-) where {package,symmetric,hermitian}
+) where {package,symmetric,hermitian,posdef,keep_input_type}
     T = Base.promote_eltype(x, y, c)
     (; conditions, backends, prep_Aᵀ) = implicit
     actual_backend = isnothing(backends) ? suggested_backend : backends.y
@@ -164,7 +171,13 @@ function build_Aᵀ_aux(
     prod! = VJP!(f_vec, prep_Aᵀ_same, actual_backend, y_vec, dc_vec, contexts)
     if package == :LinearOperators
         return LinearOperator(
-            T, length(y), length(c), symmetric, hermitian, prod!; S=typeof(dc_vec)
+            T,
+            length(y),
+            length(c),
+            symmetric,
+            hermitian,
+            prod!;
+            S=keep_input_type ? typeof(dc_vec) : Vector{T},
         )
     elseif package == :LinearMaps
         return FunctionMap{T}(
@@ -174,6 +187,7 @@ function build_Aᵀ_aux(
             ismutating=true,
             issymmetric=symmetric,
             ishermitian=hermitian,
+            isposdef=posdef,
         )
     end
 end
diff --git a/src/settings.jl b/src/settings.jl
index ca791114..5c4ba95a 100644
--- a/src/settings.jl
+++ b/src/settings.jl
@@ -12,7 +12,7 @@ Callable object that can solve linear systems `Ax = b` and `AX = B` in the same
 
 The type parameter `package` can be either:
 
-- `:Krylov` to use the solver `gmres` from [Krylov.jl](https://github.com/JuliaSmoothOptimizers/Krylov.jl) (the default)
+- `:Krylov` to use the solver `gmres` or `block_gmres` from [Krylov.jl](https://github.com/JuliaSmoothOptimizers/Krylov.jl) (the default)
 - `:IterativeSolvers` to use the solver `gmres` from [IterativeSolvers.jl](https://github.com/JuliaLinearAlgebra/IterativeSolvers.jl)
 
 Keyword arguments are passed on to the respective solver.
@@ -94,36 +94,46 @@ Specify that the matrix `A` involved in the implicit function theorem should be
 
 # Constructors
 
-    OperatorRepresentation(; symmetric=false, hermitian=false)
-    OperatorRepresentation{package}(; symmetric=false, hermitian=false)
+    OperatorRepresentation(;
+        symmetric=false, hermitian=false, posdef=false, keep_input_type=false
+    )
+    OperatorRepresentation{package}(;
+        symmetric=false, hermitian=false, posdef=false, keep_input_type=false
+    )
 
 The type parameter `package` can be either:
 
 - `:LinearOperators` to use a wrapper from [LinearOperators.jl](https://github.com/JuliaSmoothOptimizers/LinearOperators.jl) (the default)
 - `:LinearMaps` to use a wrapper from [LinearMaps.jl](https://github.com/JuliaLinearAlgebra/LinearMaps.jl)
 
-The keyword arguments `symmetric` and `hermitian` give additional properties of the Jacobian of the `conditions` with respect to the solution `y`, in case you can prove them.
+The keyword arguments `symmetric`, `hermitian` and `posdef` give additional properties of the Jacobian of the `conditions` with respect to the solution `y`, which are useful to the solver in case you can prove them.
+
+The keyword argument `keep_input_type` dictates whether to force the linear operator to work with the provided input type, or fall back on a default.
 
 # See also
 
 - [`ImplicitFunction`](@ref)
 - [`MatrixRepresentation`](@ref)
 """
-struct OperatorRepresentation{package,symmetric,hermitian} <: AbstractRepresentation
+struct OperatorRepresentation{package,symmetric,hermitian,posdef,keep_input_type} <:
+       AbstractRepresentation
     function OperatorRepresentation{package}(;
-        symmetric::Bool=false, hermitian::Bool=false
+        symmetric::Bool=false,
+        hermitian::Bool=false,
+        posdef::Bool=false,
+        keep_input_type::Bool=false,
     ) where {package}
         @assert package in [:LinearOperators, :LinearMaps]
-        return new{package,symmetric,hermitian}()
+        return new{package,symmetric,hermitian,posdef,keep_input_type}()
     end
 end
 
 function Base.show(
-    io::IO, ::OperatorRepresentation{package,symmetric,hermitian}
-) where {package,symmetric,hermitian}
+    io::IO, ::OperatorRepresentation{package,symmetric,hermitian,posdef,keep_input_type}
+) where {package,symmetric,hermitian,posdef,keep_input_type}
     return print(
         io,
-        "OperatorRepresentation{$(repr(package))}(; symmetric=$symmetric, hermitian=$hermitian)",
+        "OperatorRepresentation{$(repr(package))}(; symmetric=$symmetric, hermitian=$hermitian, posdef=$posdef, keep_input_type=$keep_input_type)",
     )
 end