diff --git a/base/linalg/special.jl b/base/linalg/special.jl
index f70d67130807b..8cd9febbdfb13 100644
--- a/base/linalg/special.jl
+++ b/base/linalg/special.jl
@@ -11,16 +11,18 @@ convert(::Type{UpperTriangular}, A::Bidiagonal) = A.isupper ? UpperTriangular(fu
 
 function convert(::Type{UnitUpperTriangular}, A::Diagonal)
     if !all(A.diag .== one(eltype(A)))
-        throw(ArgumentError("matrix cannot be represented as UnitUpperTriangular"))
+        throw(ArgumentError(string("Diagonal matrices with non-one entries on the ",
+            "diagonal cannot be converted to UnitUpperTriangular")))
     end
-    UnitUpperTriangular(full(A))
+    UnitUpperTriangular(A)
 end
 
 function convert(::Type{UnitLowerTriangular}, A::Diagonal)
     if !all(A.diag .== one(eltype(A)))
-        throw(ArgumentError("matrix cannot be represented as UnitLowerTriangular"))
+        throw(ArgumentError(string("Diagonal matrices with non-one entries on the ",
+            "diagonal cannot be converted to UnitLowerTriangular")))
     end
-    UnitLowerTriangular(full(A))
+    UnitLowerTriangular(A)
 end
 
 function convert(::Type{Diagonal}, A::Union{Bidiagonal, SymTridiagonal})
diff --git a/test/linalg/special.jl b/test/linalg/special.jl
index 89ddde5ee4b00..c7b6e1aba59c6 100644
--- a/test/linalg/special.jl
+++ b/test/linalg/special.jl
@@ -128,3 +128,13 @@ for typ in [UpperTriangular,LowerTriangular,Base.LinAlg.UnitUpperTriangular,Base
     @test Base.LinAlg.A_mul_Bc(atri,qrb[:Q]) ≈ full(atri) * qrb[:Q]'
     @test Base.LinAlg.A_mul_Bc!(copy(atri),qrb[:Q]) ≈ full(atri) * qrb[:Q]'
 end
+
+# Test conversion from Diagonal to <:AbstractTriangular
+let
+    unitdiagmat = Diagonal(ones(Int, 3))
+    # test that conversion from diagonal to triangular preserves diagonal storage structure
+    @test typeof(convert(LowerTriangular, unitdiagmat)) == LowerTriangular{Int,Diagonal{Int}}
+    @test typeof(convert(UpperTriangular, unitdiagmat)) == UpperTriangular{Int,Diagonal{Int}}
+    @test typeof(convert(Base.LinAlg.UnitLowerTriangular, unitdiagmat)) == Base.LinAlg.UnitLowerTriangular{Int,Diagonal{Int}}
+    @test typeof(convert(Base.LinAlg.UnitUpperTriangular, unitdiagmat)) == Base.LinAlg.UnitUpperTriangular{Int,Diagonal{Int}}
+end