diff --git a/src/basic.jl b/src/basic.jl
index d137131e..7501b7b0 100644
--- a/src/basic.jl
+++ b/src/basic.jl
@@ -15,6 +15,7 @@ function Base.one(L::AbstractSciMLOperator)
 end
 
 Base.convert(::Type{AbstractMatrix}, ii::IdentityOperator) = Diagonal(ones(Bool, ii.len))
+has_concretization(::IdentityOperator) = true
 
 # Copy method to avoid aliasing - IdentityOperator has no mutable fields, can return self
 Base.copy(L::IdentityOperator) = L
@@ -138,6 +139,7 @@ function Base.zero(L::AbstractSciMLOperator)
 end
 
 Base.convert(::Type{AbstractMatrix}, nn::NullOperator) = Diagonal(zeros(Bool, nn.len))
+has_concretization(::NullOperator) = true
 
 # Copy method to avoid aliasing - NullOperator has no mutable fields, can return self
 Base.copy(L::NullOperator) = L
@@ -303,6 +305,7 @@ end
 function Base.convert(::Type{AbstractMatrix}, L::ScaledOperator)
     return convert(Number, L.λ) * convert(AbstractMatrix, L.L)
 end
+has_concretization(L::ScaledOperator) = has_concretization(L.λ) & has_concretization(L.L)
 
 # traits
 function Base.show(io::IO, L::ScaledOperator{T}) where {T}
@@ -568,6 +571,7 @@ end
 function Base.convert(::Type{AbstractMatrix}, L::AddedOperator)
     return sum(op -> convert(AbstractMatrix, op), L.ops)
 end
+has_concretization(L::AddedOperator) = all(has_concretization, L.ops)
 
 # traits
 function Base.show(io::IO, L::AddedOperator)
@@ -808,6 +812,7 @@ end
 function Base.convert(::Type{AbstractMatrix}, L::ComposedOperator)
     return prod(op -> convert(AbstractMatrix, op), L.ops)
 end
+has_concretization(L::ComposedOperator) = all(has_concretization, L.ops)
 
 # traits
 function Base.show(io::IO, L::ComposedOperator)
@@ -1150,6 +1155,7 @@ Base.:/(A::AbstractSciMLOperator, B::AbstractSciMLOperator) = A * inv(B)
 function Base.convert(::Type{AbstractMatrix}, L::InvertedOperator)
     return inv(convert(AbstractMatrix, L.L))
 end
+has_concretization(L::InvertedOperator) = has_concretization(L.L)
 
 function Base.show(io::IO, L::InvertedOperator)
     print(io, "1 / ")
diff --git a/src/func.jl b/src/func.jl
index bfe37f12..c4fa023d 100644
--- a/src/func.jl
+++ b/src/func.jl
@@ -765,6 +765,7 @@ end
 
 islinear(L::FunctionOperator) = L.traits.islinear
 isconvertible(L::FunctionOperator) = L.traits.isconvertible
+has_concretization(L::FunctionOperator) = isconvertible(L)
 isconstant(L::FunctionOperator) = L.traits.isconstant
 has_adjoint(L::FunctionOperator) = !(L.op_adjoint isa Nothing)
 has_mul(::FunctionOperator{iip}) where {iip} = true
diff --git a/src/interface.jl b/src/interface.jl
index 4cc687e7..2801dc70 100644
--- a/src/interface.jl
+++ b/src/interface.jl
@@ -465,6 +465,7 @@ for pred in (
         :isposdef,
     )
     @eval function LinearAlgebra.$pred(L::AbstractSciMLOperator)
+        has_concretization(L) || return false
         if !isconvertible(L)
             @warn """using convert-based fallback in $($pred)."""
         end
diff --git a/src/left.jl b/src/left.jl
index 584b8a6e..1366107b 100644
--- a/src/left.jl
+++ b/src/left.jl
@@ -110,6 +110,7 @@ for (op, LType, VType) in (
             L.L
         )
     )
+    @eval has_concretization(L::$LType) = has_concretization(L.L)
 
     # traits
     @eval Base.size(L::$LType) = size(L.L) |> reverse
diff --git a/src/matrix.jl b/src/matrix.jl
index a2d52292..1ddd27a7 100644
--- a/src/matrix.jl
+++ b/src/matrix.jl
@@ -447,6 +447,7 @@ function Base.convert(
 end
 
 Base.convert(::Type{AbstractMatrix}, L::InvertibleOperator) = convert(AbstractMatrix, L.L)
+has_concretization(L::InvertibleOperator) = has_concretization(L.L)
 
 # traits
 function Base.show(io::IO, L::InvertibleOperator)
diff --git a/src/scalar.jl b/src/scalar.jl
index 7dea75fe..8abc15ed 100644
--- a/src/scalar.jl
+++ b/src/scalar.jl
@@ -35,6 +35,7 @@ has_mul!(::AbstractSciMLScalarOperator) = true
 isconcrete(::AbstractSciMLScalarOperator) = true
 islinear(::AbstractSciMLScalarOperator) = true
 has_adjoint(::AbstractSciMLScalarOperator) = true
+has_concretization(::AbstractSciMLScalarOperator) = true
 
 Base.:*(α::AbstractSciMLScalarOperator, u::AbstractArray) = convert(Number, α) * u
 Base.:\(α::AbstractSciMLScalarOperator, u::AbstractArray) = convert(Number, α) \ u
diff --git a/src/tensor.jl b/src/tensor.jl
index 9a0a0ed8..89cef769 100644
--- a/src/tensor.jl
+++ b/src/tensor.jl
@@ -180,6 +180,7 @@ end
 
 islinear(L::TensorProductOperator) = reduce(&, islinear.(L.ops))
 isconvertible(::TensorProductOperator) = false
+has_concretization(L::TensorProductOperator) = all(has_concretization, L.ops)
 Base.iszero(L::TensorProductOperator) = reduce(|, iszero.(L.ops))
 has_adjoint(L::TensorProductOperator) = reduce(&, has_adjoint.(L.ops))
 has_mul(L::TensorProductOperator) = reduce(&, has_mul.(L.ops))
diff --git a/src/utils.jl b/src/utils.jl
index 01a38d46..1506122e 100644
--- a/src/utils.jl
+++ b/src/utils.jl
@@ -81,3 +81,4 @@ _unwrap_val(x) = x
 _unwrap_val(::Val{X}) where {X} = X
 
 has_concretization(::AbstractSciMLOperator) = false
+has_concretization(::Union{AbstractMatrix, UniformScaling, Factorization, Number}) = true
diff --git a/test/basic.jl b/test/basic.jl
index dac104b8..6ad4a6bf 100644
--- a/test/basic.jl
+++ b/test/basic.jl
@@ -420,6 +420,29 @@ end
     @test ldiv!(rand(N), op, u) ≈ op \ u
 end
 
+@testset "has_concretization composites" begin
+    A = MatrixOperator(rand(N, N))
+    B = MatrixOperator(rand(N, N))
+    F = FunctionOperator(
+        (du, u, p, t) -> copyto!(du, u),
+        zeros(N),
+        zeros(N);
+        isinplace = true,
+        T = Float64,
+        islinear = true
+    )
+
+    @test has_concretization(A)
+    @test has_concretization(2A)
+    @test has_concretization(A + B)
+    @test has_concretization(A * B)
+    @test has_concretization(inv(A))
+    @test !has_concretization(F)
+    @test !has_concretization(F * A)
+    @test !has_concretization(A + F)
+    @test !ishermitian(F * A)
+end
+
 @testset "Adjoint, Transpose" begin
     for (
             op,