diff --git a/stdlib/LinearAlgebra/src/cholesky.jl b/stdlib/LinearAlgebra/src/cholesky.jl
index 545d92ec1704d..03f7c273ccbef 100644
--- a/stdlib/LinearAlgebra/src/cholesky.jl
+++ b/stdlib/LinearAlgebra/src/cholesky.jl
@@ -631,11 +631,16 @@ function Cholesky{T}(C::Cholesky) where T
     Cnew = convert(AbstractMatrix{T}, C.factors)
     Cholesky{T, typeof(Cnew)}(Cnew, C.uplo, C.info)
 end
+Cholesky{T,S}(C::Cholesky) where {T,S<:AbstractMatrix} = Cholesky{T,S}(C.factors, C.uplo, C.info)
 Factorization{T}(C::Cholesky{T}) where {T} = C
 Factorization{T}(C::Cholesky) where {T} = Cholesky{T}(C)
 CholeskyPivoted{T}(C::CholeskyPivoted{T}) where {T} = C
 CholeskyPivoted{T}(C::CholeskyPivoted) where {T} =
-    CholeskyPivoted(AbstractMatrix{T}(C.factors),C.uplo,C.piv,C.rank,C.tol,C.info)
+    CholeskyPivoted(AbstractMatrix{T}(C.factors), C.uplo, C.piv, C.rank, C.tol, C.info)
+CholeskyPivoted{T,S}(C::CholeskyPivoted) where {T,S<:AbstractMatrix} =
+    CholeskyPivoted{T,S,typeof(C.piv)}(C.factors, C.uplo, C.piv, C.rank, C.tol, C.info)
+CholeskyPivoted{T,S,P}(C::CholeskyPivoted) where {T,S<:AbstractMatrix,P<:AbstractVector{<:Integer}} =
+    CholeskyPivoted{T,S,P}(C.factors, C.uplo, C.piv, C.rank, C.tol, C.info)
 Factorization{T}(C::CholeskyPivoted{T}) where {T} = C
 Factorization{T}(C::CholeskyPivoted) where {T} = CholeskyPivoted{T}(C)
 
diff --git a/stdlib/LinearAlgebra/test/cholesky.jl b/stdlib/LinearAlgebra/test/cholesky.jl
index 00bfc18a21638..6ba72432048a9 100644
--- a/stdlib/LinearAlgebra/test/cholesky.jl
+++ b/stdlib/LinearAlgebra/test/cholesky.jl
@@ -281,6 +281,24 @@ end
             end
         end
     end
+
+    @testset "eltype/matrixtype conversions" begin
+        apd  = Matrix(Hermitian(areal'*areal))
+        capd = cholesky(apd)
+        @test convert(Cholesky{Float64}, capd) === capd
+        @test convert(Cholesky{Float64,Matrix{Float64}}, capd) === convert(typeof(capd), capd) === capd
+        @test eltype(convert(Cholesky{Float32}, capd)) === Float32
+        @test eltype(convert(Cholesky{Float32,Matrix{Float32}}, capd)) === Float32
+
+        capd = cholesky(apd, RowMaximum())
+        @test convert(CholeskyPivoted{Float64}, capd) === capd
+        @test convert(CholeskyPivoted{Float64,Matrix{Float64}}, capd) === capd
+        @test convert(CholeskyPivoted{Float64,Matrix{Float64},Vector{Int}}, capd) === convert(typeof(capd), capd) === capd
+        @test eltype(convert(CholeskyPivoted{Float32}, capd)) === Float32
+        @test eltype(convert(CholeskyPivoted{Float32,Matrix{Float32}}, capd)) === Float32
+        @test eltype(convert(CholeskyPivoted{Float32,Matrix{Float32},Vector{Int}}, capd)) === Float32
+        @test eltype(convert(CholeskyPivoted{Float32,Matrix{Float32},Vector{Int16}}, capd).piv) === Int16
+    end
 end
 
 @testset "behavior for non-positive definite matrices" for T in (Float64, ComplexF64, BigFloat)