diff --git a/Eigen/src/Cholesky/LDLT.h b/Eigen/src/Cholesky/LDLT.h
index f47b2ea56..a19e947a4 100644
--- a/Eigen/src/Cholesky/LDLT.h
+++ b/Eigen/src/Cholesky/LDLT.h
@@ -331,16 +331,16 @@ template<> struct ldlt_inplace<Upper>
 
 template<typename MatrixType> struct LDLT_Traits<MatrixType,Lower>
 {
-  typedef TriangularView<MatrixType, UnitLower> MatrixL;
-  typedef TriangularView<typename MatrixType::AdjointReturnType, UnitUpper> MatrixU;
+  typedef const TriangularView<const MatrixType, UnitLower> MatrixL;
+  typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitUpper> MatrixU;
   inline static MatrixL getL(const MatrixType& m) { return m; }
   inline static MatrixU getU(const MatrixType& m) { return m.adjoint(); }
 };
 
 template<typename MatrixType> struct LDLT_Traits<MatrixType,Upper>
 {
-  typedef TriangularView<typename MatrixType::AdjointReturnType, UnitLower> MatrixL;
-  typedef TriangularView<MatrixType, UnitUpper> MatrixU;
+  typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitLower> MatrixL;
+  typedef const TriangularView<const MatrixType, UnitUpper> MatrixU;
   inline static MatrixL getL(const MatrixType& m) { return m.adjoint(); }
   inline static MatrixU getU(const MatrixType& m) { return m; }
 };
diff --git a/Eigen/src/Cholesky/LLT.h b/Eigen/src/Cholesky/LLT.h
index a4ee5b11c..3bb76b578 100644
--- a/Eigen/src/Cholesky/LLT.h
+++ b/Eigen/src/Cholesky/LLT.h
@@ -274,8 +274,8 @@ template<> struct llt_inplace<Upper>
 
 template<typename MatrixType> struct LLT_Traits<MatrixType,Lower>
 {
-  typedef TriangularView<MatrixType, Lower> MatrixL;
-  typedef TriangularView<typename MatrixType::AdjointReturnType, Upper> MatrixU;
+  typedef const TriangularView<const MatrixType, Lower> MatrixL;
+  typedef const TriangularView<const typename MatrixType::AdjointReturnType, Upper> MatrixU;
   inline static MatrixL getL(const MatrixType& m) { return m; }
   inline static MatrixU getU(const MatrixType& m) { return m.adjoint(); }
   static bool inplace_decomposition(MatrixType& m)
@@ -284,8 +284,8 @@ template<typename MatrixType> struct LLT_Traits<MatrixType,Lower>
 
 template<typename MatrixType> struct LLT_Traits<MatrixType,Upper>
 {
-  typedef TriangularView<typename MatrixType::AdjointReturnType, Lower> MatrixL;
-  typedef TriangularView<MatrixType, Upper> MatrixU;
+  typedef const TriangularView<const typename MatrixType::AdjointReturnType, Lower> MatrixL;
+  typedef const TriangularView<const MatrixType, Upper> MatrixU;
   inline static MatrixL getL(const MatrixType& m) { return m.adjoint(); }
   inline static MatrixU getU(const MatrixType& m) { return m; }
   static bool inplace_decomposition(MatrixType& m)
diff --git a/test/cholesky.cpp b/test/cholesky.cpp
index ae4342cee..16a01ce4f 100644
--- a/test/cholesky.cpp
+++ b/test/cholesky.cpp
@@ -124,6 +124,11 @@ template<typename MatrixType> void cholesky(const MatrixType& m)
     MatrixType neg = -symmLo;
     chollo.compute(neg);
     VERIFY(chollo.info()==NumericalIssue);
+
+    VERIFY_IS_APPROX(MatrixType(chollo.matrixL().transpose().conjugate()), MatrixType(chollo.matrixU()));
+    VERIFY_IS_APPROX(MatrixType(chollo.matrixU().transpose().conjugate()), MatrixType(chollo.matrixL()));
+    VERIFY_IS_APPROX(MatrixType(cholup.matrixL().transpose().conjugate()), MatrixType(cholup.matrixU()));
+    VERIFY_IS_APPROX(MatrixType(cholup.matrixU().transpose().conjugate()), MatrixType(cholup.matrixL()));
   }
 
   // LDLT
@@ -152,6 +157,11 @@ template<typename MatrixType> void cholesky(const MatrixType& m)
     matX = ldltup.solve(matB);
     VERIFY_IS_APPROX(symm * matX, matB);
 
+    VERIFY_IS_APPROX(MatrixType(ldltlo.matrixL().transpose().conjugate()), MatrixType(ldltlo.matrixU()));
+    VERIFY_IS_APPROX(MatrixType(ldltlo.matrixU().transpose().conjugate()), MatrixType(ldltlo.matrixL()));
+    VERIFY_IS_APPROX(MatrixType(ldltup.matrixL().transpose().conjugate()), MatrixType(ldltup.matrixU()));
+    VERIFY_IS_APPROX(MatrixType(ldltup.matrixU().transpose().conjugate()), MatrixType(ldltup.matrixL()));
+
     if(MatrixType::RowsAtCompileTime==Dynamic)
     {
       // note : each inplace permutation requires a small temporary vector (mask)