diff --git a/Eigen/src/SparseCholesky/SimplicialCholesky.h b/Eigen/src/SparseCholesky/SimplicialCholesky.h
index 423287b5b..f3ce975cf 100644
--- a/Eigen/src/SparseCholesky/SimplicialCholesky.h
+++ b/Eigen/src/SparseCholesky/SimplicialCholesky.h
@@ -58,6 +58,7 @@ class SimplicialCholeskyBase : public SparseSolverBase<Derived> {
   enum { UpLo = internal::traits<Derived>::UpLo };
   typedef typename MatrixType::Scalar Scalar;
   typedef typename MatrixType::RealScalar RealScalar;
+  typedef typename internal::traits<Derived>::DiagonalScalar DiagonalScalar;
   typedef typename MatrixType::StorageIndex StorageIndex;
   typedef SparseMatrix<Scalar, ColMajor, StorageIndex> CholMatrixType;
   typedef CholMatrixType const* ConstCholMatrixPtr;
@@ -114,7 +115,7 @@ class SimplicialCholeskyBase : public SparseSolverBase<Derived> {
    *
    * \returns a reference to \c *this.
    */
-  Derived& setShift(const RealScalar& offset, const RealScalar& scale = 1) {
+  Derived& setShift(const DiagonalScalar& offset, const DiagonalScalar& scale = 1) {
     m_shiftOffset = offset;
     m_shiftScale = scale;
     return derived();
@@ -178,18 +179,18 @@ class SimplicialCholeskyBase : public SparseSolverBase<Derived> {
 
  protected:
   /** Computes the sparse Cholesky decomposition of \a matrix */
-  template <bool DoLDLT>
+  template <bool DoLDLT, bool NonHermitian>
   void compute(const MatrixType& matrix) {
     eigen_assert(matrix.rows() == matrix.cols());
     Index size = matrix.cols();
     CholMatrixType tmp(size, size);
     ConstCholMatrixPtr pmat;
-    ordering(matrix, pmat, tmp);
+    ordering<NonHermitian>(matrix, pmat, tmp);
     analyzePattern_preordered(*pmat, DoLDLT);
-    factorize_preordered<DoLDLT>(*pmat);
+    factorize_preordered<DoLDLT, NonHermitian>(*pmat);
   }
 
-  template <bool DoLDLT>
+  template <bool DoLDLT, bool NonHermitian>
   void factorize(const MatrixType& a) {
     eigen_assert(a.rows() == a.cols());
     Index size = a.cols();
@@ -200,28 +201,33 @@ class SimplicialCholeskyBase : public SparseSolverBase<Derived> {
       // If there is no ordering, try to directly use the input matrix without any copy
       internal::simplicial_cholesky_grab_input<CholMatrixType, MatrixType>::run(a, pmat, tmp);
     } else {
-      tmp.template selfadjointView<Upper>() = a.template selfadjointView<UpLo>().twistedBy(m_P);
+      internal::permute_symm_to_symm<UpLo, Upper, NonHermitian>(a, tmp, m_P.indices().data());
       pmat = &tmp;
     }
 
-    factorize_preordered<DoLDLT>(*pmat);
+    factorize_preordered<DoLDLT, NonHermitian>(*pmat);
   }
 
-  template <bool DoLDLT>
+  template <bool DoLDLT, bool NonHermitian>
   void factorize_preordered(const CholMatrixType& a);
 
-  void analyzePattern(const MatrixType& a, bool doLDLT) {
+  template <bool DoLDLT, bool NonHermitian>
+  void analyzePattern(const MatrixType& a) {
     eigen_assert(a.rows() == a.cols());
     Index size = a.cols();
     CholMatrixType tmp(size, size);
     ConstCholMatrixPtr pmat;
-    ordering(a, pmat, tmp);
-    analyzePattern_preordered(*pmat, doLDLT);
+    ordering<NonHermitian>(a, pmat, tmp);
+    analyzePattern_preordered(*pmat, DoLDLT);
   }
   void analyzePattern_preordered(const CholMatrixType& a, bool doLDLT);
 
+  template <bool NonHermitian>
   void ordering(const MatrixType& a, ConstCholMatrixPtr& pmat, CholMatrixType& ap);
 
+  inline DiagonalScalar getDiag(Scalar x) { return internal::traits<Derived>::getDiag(x); }
+  inline Scalar getSymm(Scalar x) { return internal::traits<Derived>::getSymm(x); }
+
   /** keeps off-diagonal entries; drops diagonal entries */
   struct keep_diag {
     inline bool operator()(const Index& row, const Index& col, const Scalar&) const { return row != col; }
@@ -238,8 +244,8 @@ class SimplicialCholeskyBase : public SparseSolverBase<Derived> {
   PermutationMatrix<Dynamic, Dynamic, StorageIndex> m_P;     // the permutation
   PermutationMatrix<Dynamic, Dynamic, StorageIndex> m_Pinv;  // the inverse permutation
 
-  RealScalar m_shiftOffset;
-  RealScalar m_shiftScale;
+  DiagonalScalar m_shiftOffset;
+  DiagonalScalar m_shiftScale;
 };
 
 template <typename MatrixType_, int UpLo_ = Lower,
@@ -248,6 +254,12 @@ class SimplicialLLT;
 template <typename MatrixType_, int UpLo_ = Lower,
           typename Ordering_ = AMDOrdering<typename MatrixType_::StorageIndex> >
 class SimplicialLDLT;
+template <typename MatrixType_, int UpLo_ = Lower,
+          typename Ordering_ = AMDOrdering<typename MatrixType_::StorageIndex> >
+class SimplicialNonHermitianLLT;
+template <typename MatrixType_, int UpLo_ = Lower,
+          typename Ordering_ = AMDOrdering<typename MatrixType_::StorageIndex> >
+class SimplicialNonHermitianLDLT;
 template <typename MatrixType_, int UpLo_ = Lower,
           typename Ordering_ = AMDOrdering<typename MatrixType_::StorageIndex> >
 class SimplicialCholesky;
@@ -260,12 +272,15 @@ struct traits<SimplicialLLT<MatrixType_, UpLo_, Ordering_> > {
   typedef Ordering_ OrderingType;
   enum { UpLo = UpLo_ };
   typedef typename MatrixType::Scalar Scalar;
+  typedef typename MatrixType::RealScalar DiagonalScalar;
   typedef typename MatrixType::StorageIndex StorageIndex;
   typedef SparseMatrix<Scalar, ColMajor, StorageIndex> CholMatrixType;
   typedef TriangularView<const CholMatrixType, Eigen::Lower> MatrixL;
   typedef TriangularView<const typename CholMatrixType::AdjointReturnType, Eigen::Upper> MatrixU;
   static inline MatrixL getL(const CholMatrixType& m) { return MatrixL(m); }
   static inline MatrixU getU(const CholMatrixType& m) { return MatrixU(m.adjoint()); }
+  static inline DiagonalScalar getDiag(Scalar x) { return numext::real(x); }
+  static inline Scalar getSymm(Scalar x) { return numext::conj(x); }
 };
 
 template <typename MatrixType_, int UpLo_, typename Ordering_>
@@ -274,12 +289,49 @@ struct traits<SimplicialLDLT<MatrixType_, UpLo_, Ordering_> > {
   typedef Ordering_ OrderingType;
   enum { UpLo = UpLo_ };
   typedef typename MatrixType::Scalar Scalar;
+  typedef typename MatrixType::RealScalar DiagonalScalar;
   typedef typename MatrixType::StorageIndex StorageIndex;
   typedef SparseMatrix<Scalar, ColMajor, StorageIndex> CholMatrixType;
   typedef TriangularView<const CholMatrixType, Eigen::UnitLower> MatrixL;
   typedef TriangularView<const typename CholMatrixType::AdjointReturnType, Eigen::UnitUpper> MatrixU;
   static inline MatrixL getL(const CholMatrixType& m) { return MatrixL(m); }
   static inline MatrixU getU(const CholMatrixType& m) { return MatrixU(m.adjoint()); }
+  static inline DiagonalScalar getDiag(Scalar x) { return numext::real(x); }
+  static inline Scalar getSymm(Scalar x) { return numext::conj(x); }
+};
+
+template <typename MatrixType_, int UpLo_, typename Ordering_>
+struct traits<SimplicialNonHermitianLLT<MatrixType_, UpLo_, Ordering_> > {
+  typedef MatrixType_ MatrixType;
+  typedef Ordering_ OrderingType;
+  enum { UpLo = UpLo_ };
+  typedef typename MatrixType::Scalar Scalar;
+  typedef typename MatrixType::Scalar DiagonalScalar;
+  typedef typename MatrixType::StorageIndex StorageIndex;
+  typedef SparseMatrix<Scalar, ColMajor, StorageIndex> CholMatrixType;
+  typedef TriangularView<const CholMatrixType, Eigen::Lower> MatrixL;
+  typedef TriangularView<const typename CholMatrixType::ConstTransposeReturnType, Eigen::Upper> MatrixU;
+  static inline MatrixL getL(const CholMatrixType& m) { return MatrixL(m); }
+  static inline MatrixU getU(const CholMatrixType& m) { return MatrixU(m.transpose()); }
+  static inline DiagonalScalar getDiag(Scalar x) { return x; }
+  static inline Scalar getSymm(Scalar x) { return x; }
+};
+
+template <typename MatrixType_, int UpLo_, typename Ordering_>
+struct traits<SimplicialNonHermitianLDLT<MatrixType_, UpLo_, Ordering_> > {
+  typedef MatrixType_ MatrixType;
+  typedef Ordering_ OrderingType;
+  enum { UpLo = UpLo_ };
+  typedef typename MatrixType::Scalar Scalar;
+  typedef typename MatrixType::Scalar DiagonalScalar;
+  typedef typename MatrixType::StorageIndex StorageIndex;
+  typedef SparseMatrix<Scalar, ColMajor, StorageIndex> CholMatrixType;
+  typedef TriangularView<const CholMatrixType, Eigen::UnitLower> MatrixL;
+  typedef TriangularView<const typename CholMatrixType::ConstTransposeReturnType, Eigen::UnitUpper> MatrixU;
+  static inline MatrixL getL(const CholMatrixType& m) { return MatrixL(m); }
+  static inline MatrixU getU(const CholMatrixType& m) { return MatrixU(m.transpose()); }
+  static inline DiagonalScalar getDiag(Scalar x) { return x; }
+  static inline Scalar getSymm(Scalar x) { return x; }
 };
 
 template <typename MatrixType_, int UpLo_, typename Ordering_>
@@ -287,6 +339,10 @@ struct traits<SimplicialCholesky<MatrixType_, UpLo_, Ordering_> > {
   typedef MatrixType_ MatrixType;
   typedef Ordering_ OrderingType;
   enum { UpLo = UpLo_ };
+  typedef typename MatrixType::Scalar Scalar;
+  typedef typename MatrixType::RealScalar DiagonalScalar;
+  static inline DiagonalScalar getDiag(Scalar x) { return numext::real(x); }
+  static inline Scalar getSymm(Scalar x) { return numext::conj(x); }
 };
 
 }  // namespace internal
@@ -346,7 +402,7 @@ class SimplicialLLT : public SimplicialCholeskyBase<SimplicialLLT<MatrixType_, U
 
   /** Computes the sparse Cholesky decomposition of \a matrix */
   SimplicialLLT& compute(const MatrixType& matrix) {
-    Base::template compute<false>(matrix);
+    Base::template compute<false, false>(matrix);
     return *this;
   }
 
@@ -356,7 +412,7 @@ class SimplicialLLT : public SimplicialCholeskyBase<SimplicialLLT<MatrixType_, U
    *
    * \sa factorize()
    */
-  void analyzePattern(const MatrixType& a) { Base::analyzePattern(a, false); }
+  void analyzePattern(const MatrixType& a) { Base::template analyzePattern<false, false>(a); }
 
   /** Performs a numeric decomposition of \a matrix
    *
@@ -364,7 +420,7 @@ class SimplicialLLT : public SimplicialCholeskyBase<SimplicialLLT<MatrixType_, U
    *
    * \sa analyzePattern()
    */
-  void factorize(const MatrixType& a) { Base::template factorize<false>(a); }
+  void factorize(const MatrixType& a) { Base::template factorize<false, false>(a); }
 
   /** \returns the determinant of the underlying matrix from the current factorization */
   Scalar determinant() const {
@@ -434,7 +490,7 @@ class SimplicialLDLT : public SimplicialCholeskyBase<SimplicialLDLT<MatrixType_,
 
   /** Computes the sparse Cholesky decomposition of \a matrix */
   SimplicialLDLT& compute(const MatrixType& matrix) {
-    Base::template compute<true>(matrix);
+    Base::template compute<true, false>(matrix);
     return *this;
   }
 
@@ -444,7 +500,7 @@ class SimplicialLDLT : public SimplicialCholeskyBase<SimplicialLDLT<MatrixType_,
    *
    * \sa factorize()
    */
-  void analyzePattern(const MatrixType& a) { Base::analyzePattern(a, true); }
+  void analyzePattern(const MatrixType& a) { Base::template analyzePattern<true, false>(a); }
 
   /** Performs a numeric decomposition of \a matrix
    *
@@ -452,7 +508,177 @@ class SimplicialLDLT : public SimplicialCholeskyBase<SimplicialLDLT<MatrixType_,
    *
    * \sa analyzePattern()
    */
-  void factorize(const MatrixType& a) { Base::template factorize<true>(a); }
+  void factorize(const MatrixType& a) { Base::template factorize<true, false>(a); }
+
+  /** \returns the determinant of the underlying matrix from the current factorization */
+  Scalar determinant() const { return Base::m_diag.prod(); }
+};
+
+/** \ingroup SparseCholesky_Module
+ * \class SimplicialNonHermitianLLT
+ * \brief A direct sparse LLT Cholesky factorizations, for symmetric non-hermitian matrices.
+ *
+ * This class provides a LL^T Cholesky factorizations of sparse matrices that are
+ * symmetric but not hermitian. For real matrices, this is equivalent to the regular LLT factorization.
+ * The factorization allows for solving A.X = B where X and B can be either dense or sparse.
+ *
+ * In order to reduce the fill-in, a symmetric permutation P is applied prior to the factorization
+ * such that the factorized matrix is P A P^-1.
+ *
+ * \tparam MatrixType_ the type of the sparse matrix A, it must be a SparseMatrix<>
+ * \tparam UpLo_ the triangular part that will be used for the computations. It can be Lower
+ *               or Upper. Default is Lower.
+ * \tparam Ordering_ The ordering method to use, either AMDOrdering<> or NaturalOrdering<>. Default is AMDOrdering<>
+ *
+ * \implsparsesolverconcept
+ *
+ * \sa class SimplicialNonHermitianLDLT, SimplicialLLT, class AMDOrdering, class NaturalOrdering
+ */
+template <typename MatrixType_, int UpLo_, typename Ordering_>
+class SimplicialNonHermitianLLT
+    : public SimplicialCholeskyBase<SimplicialNonHermitianLLT<MatrixType_, UpLo_, Ordering_> > {
+ public:
+  typedef MatrixType_ MatrixType;
+  enum { UpLo = UpLo_ };
+  typedef SimplicialCholeskyBase<SimplicialNonHermitianLLT> Base;
+  typedef typename MatrixType::Scalar Scalar;
+  typedef typename MatrixType::RealScalar RealScalar;
+  typedef typename MatrixType::StorageIndex StorageIndex;
+  typedef SparseMatrix<Scalar, ColMajor, StorageIndex> CholMatrixType;
+  typedef Matrix<Scalar, Dynamic, 1> VectorType;
+  typedef internal::traits<SimplicialNonHermitianLLT> Traits;
+  typedef typename Traits::MatrixL MatrixL;
+  typedef typename Traits::MatrixU MatrixU;
+
+ public:
+  /** Default constructor */
+  SimplicialNonHermitianLLT() : Base() {}
+
+  /** Constructs and performs the LLT factorization of \a matrix */
+  explicit SimplicialNonHermitianLLT(const MatrixType& matrix) : Base(matrix) {}
+
+  /** \returns an expression of the factor L */
+  inline const MatrixL matrixL() const {
+    eigen_assert(Base::m_factorizationIsOk && "Simplicial LLT not factorized");
+    return Traits::getL(Base::m_matrix);
+  }
+
+  /** \returns an expression of the factor U (= L^*) */
+  inline const MatrixU matrixU() const {
+    eigen_assert(Base::m_factorizationIsOk && "Simplicial LLT not factorized");
+    return Traits::getU(Base::m_matrix);
+  }
+
+  /** Computes the sparse Cholesky decomposition of \a matrix */
+  SimplicialNonHermitianLLT& compute(const MatrixType& matrix) {
+    Base::template compute<false, true>(matrix);
+    return *this;
+  }
+
+  /** Performs a symbolic decomposition on the sparcity of \a matrix.
+   *
+   * This function is particularly useful when solving for several problems having the same structure.
+   *
+   * \sa factorize()
+   */
+  void analyzePattern(const MatrixType& a) { Base::template analyzePattern<false, true>(a); }
+
+  /** Performs a numeric decomposition of \a matrix
+   *
+   * The given matrix must has the same sparcity than the matrix on which the symbolic decomposition has been performed.
+   *
+   * \sa analyzePattern()
+   */
+  void factorize(const MatrixType& a) { Base::template factorize<false, true>(a); }
+
+  /** \returns the determinant of the underlying matrix from the current factorization */
+  Scalar determinant() const {
+    Scalar detL = Base::m_matrix.diagonal().prod();
+    return detL * detL;
+  }
+};
+
+/** \ingroup SparseCholesky_Module
+ * \class SimplicialNonHermitianLDLT
+ * \brief A direct sparse LDLT Cholesky factorizations without square root, for symmetric non-hermitian matrices.
+ *
+ * This class provides a LDL^T Cholesky factorizations without square root of sparse matrices that are
+ * symmetric but not hermitian. For real matrices, this is equivalent to the regular LDLT factorization.
+ * The factorization allows for solving A.X = B where X and B can be either dense or sparse.
+ *
+ * In order to reduce the fill-in, a symmetric permutation P is applied prior to the factorization
+ * such that the factorized matrix is P A P^-1.
+ *
+ * \tparam MatrixType_ the type of the sparse matrix A, it must be a SparseMatrix<>
+ * \tparam UpLo_ the triangular part that will be used for the computations. It can be Lower
+ *               or Upper. Default is Lower.
+ * \tparam Ordering_ The ordering method to use, either AMDOrdering<> or NaturalOrdering<>. Default is AMDOrdering<>
+ *
+ * \implsparsesolverconcept
+ *
+ * \sa class SimplicialNonHermitianLLT, SimplicialLDLT, class AMDOrdering, class NaturalOrdering
+ */
+template <typename MatrixType_, int UpLo_, typename Ordering_>
+class SimplicialNonHermitianLDLT
+    : public SimplicialCholeskyBase<SimplicialNonHermitianLDLT<MatrixType_, UpLo_, Ordering_> > {
+ public:
+  typedef MatrixType_ MatrixType;
+  enum { UpLo = UpLo_ };
+  typedef SimplicialCholeskyBase<SimplicialNonHermitianLDLT> Base;
+  typedef typename MatrixType::Scalar Scalar;
+  typedef typename MatrixType::RealScalar RealScalar;
+  typedef typename MatrixType::StorageIndex StorageIndex;
+  typedef SparseMatrix<Scalar, ColMajor, StorageIndex> CholMatrixType;
+  typedef Matrix<Scalar, Dynamic, 1> VectorType;
+  typedef internal::traits<SimplicialNonHermitianLDLT> Traits;
+  typedef typename Traits::MatrixL MatrixL;
+  typedef typename Traits::MatrixU MatrixU;
+
+ public:
+  /** Default constructor */
+  SimplicialNonHermitianLDLT() : Base() {}
+
+  /** Constructs and performs the LLT factorization of \a matrix */
+  explicit SimplicialNonHermitianLDLT(const MatrixType& matrix) : Base(matrix) {}
+
+  /** \returns a vector expression of the diagonal D */
+  inline const VectorType vectorD() const {
+    eigen_assert(Base::m_factorizationIsOk && "Simplicial LDLT not factorized");
+    return Base::m_diag;
+  }
+  /** \returns an expression of the factor L */
+  inline const MatrixL matrixL() const {
+    eigen_assert(Base::m_factorizationIsOk && "Simplicial LDLT not factorized");
+    return Traits::getL(Base::m_matrix);
+  }
+
+  /** \returns an expression of the factor U (= L^*) */
+  inline const MatrixU matrixU() const {
+    eigen_assert(Base::m_factorizationIsOk && "Simplicial LDLT not factorized");
+    return Traits::getU(Base::m_matrix);
+  }
+
+  /** Computes the sparse Cholesky decomposition of \a matrix */
+  SimplicialNonHermitianLDLT& compute(const MatrixType& matrix) {
+    Base::template compute<true, true>(matrix);
+    return *this;
+  }
+
+  /** Performs a symbolic decomposition on the sparcity of \a matrix.
+   *
+   * This function is particularly useful when solving for several problems having the same structure.
+   *
+   * \sa factorize()
+   */
+  void analyzePattern(const MatrixType& a) { Base::template analyzePattern<true, true>(a); }
+
+  /** Performs a numeric decomposition of \a matrix
+   *
+   * The given matrix must has the same sparcity than the matrix on which the symbolic decomposition has been performed.
+   *
+   * \sa analyzePattern()
+   */
+  void factorize(const MatrixType& a) { Base::template factorize<true, true>(a); }
 
   /** \returns the determinant of the underlying matrix from the current factorization */
   Scalar determinant() const { return Base::m_diag.prod(); }
@@ -475,7 +701,6 @@ class SimplicialCholesky : public SimplicialCholeskyBase<SimplicialCholesky<Matr
   typedef typename MatrixType::StorageIndex StorageIndex;
   typedef SparseMatrix<Scalar, ColMajor, StorageIndex> CholMatrixType;
   typedef Matrix<Scalar, Dynamic, 1> VectorType;
-  typedef internal::traits<SimplicialCholesky> Traits;
   typedef internal::traits<SimplicialLDLT<MatrixType, UpLo> > LDLTTraits;
   typedef internal::traits<SimplicialLLT<MatrixType, UpLo> > LLTTraits;
 
@@ -511,9 +736,9 @@ class SimplicialCholesky : public SimplicialCholeskyBase<SimplicialCholesky<Matr
   /** Computes the sparse Cholesky decomposition of \a matrix */
   SimplicialCholesky& compute(const MatrixType& matrix) {
     if (m_LDLT)
-      Base::template compute<true>(matrix);
+      Base::template compute<true, false>(matrix);
     else
-      Base::template compute<false>(matrix);
+      Base::template compute<false, false>(matrix);
     return *this;
   }
 
@@ -523,7 +748,12 @@ class SimplicialCholesky : public SimplicialCholeskyBase<SimplicialCholesky<Matr
    *
    * \sa factorize()
    */
-  void analyzePattern(const MatrixType& a) { Base::analyzePattern(a, m_LDLT); }
+  void analyzePattern(const MatrixType& a) {
+    if (m_LDLT)
+      Base::template analyzePattern<true, false>(a);
+    else
+      Base::template analyzePattern<false, false>(a);
+  }
 
   /** Performs a numeric decomposition of \a matrix
    *
@@ -533,9 +763,9 @@ class SimplicialCholesky : public SimplicialCholeskyBase<SimplicialCholesky<Matr
    */
   void factorize(const MatrixType& a) {
     if (m_LDLT)
-      Base::template factorize<true>(a);
+      Base::template factorize<true, false>(a);
     else
-      Base::template factorize<false>(a);
+      Base::template factorize<false, false>(a);
   }
 
   /** \internal */
@@ -594,6 +824,7 @@ class SimplicialCholesky : public SimplicialCholeskyBase<SimplicialCholesky<Matr
 };
 
 template <typename Derived>
+template <bool NonHermitian>
 void SimplicialCholeskyBase<Derived>::ordering(const MatrixType& a, ConstCholMatrixPtr& pmat, CholMatrixType& ap) {
   eigen_assert(a.rows() == a.cols());
   const Index size = a.rows();
@@ -602,7 +833,7 @@ void SimplicialCholeskyBase<Derived>::ordering(const MatrixType& a, ConstCholMat
   if (!internal::is_same<OrderingType, NaturalOrdering<Index> >::value) {
     {
       CholMatrixType C;
-      C = a.template selfadjointView<UpLo>();
+      internal::permute_symm_to_fullsymm<UpLo, NonHermitian>(a, C, NULL);
 
       OrderingType ordering;
       ordering(C, m_Pinv);
@@ -614,14 +845,14 @@ void SimplicialCholeskyBase<Derived>::ordering(const MatrixType& a, ConstCholMat
       m_P.resize(0);
 
     ap.resize(size, size);
-    ap.template selfadjointView<Upper>() = a.template selfadjointView<UpLo>().twistedBy(m_P);
+    internal::permute_symm_to_symm<UpLo, Upper, NonHermitian>(a, ap, m_P.indices().data());
   } else {
     m_Pinv.resize(0);
     m_P.resize(0);
     if (int(UpLo) == int(Lower) || MatrixType::IsRowMajor) {
       // we have to transpose the lower part to to the upper one
       ap.resize(size, size);
-      ap.template selfadjointView<Upper>() = a.template selfadjointView<UpLo>();
+      internal::permute_symm_to_symm<UpLo, Upper, NonHermitian>(a, ap, NULL);
     } else
       internal::simplicial_cholesky_grab_input<CholMatrixType, MatrixType>::run(a, pmat, ap);
   }
diff --git a/Eigen/src/SparseCholesky/SimplicialCholesky_impl.h b/Eigen/src/SparseCholesky/SimplicialCholesky_impl.h
index abfbbe6bc..0b13c56b5 100644
--- a/Eigen/src/SparseCholesky/SimplicialCholesky_impl.h
+++ b/Eigen/src/SparseCholesky/SimplicialCholesky_impl.h
@@ -67,7 +67,7 @@ void SimplicialCholeskyBase<Derived>::analyzePattern_preordered(const CholMatrix
 }
 
 template <typename Derived>
-template <bool DoLDLT>
+template <bool DoLDLT, bool NonHermitian>
 void SimplicialCholeskyBase<Derived>::factorize_preordered(const CholMatrixType& ap) {
   using std::sqrt;
 
@@ -97,7 +97,7 @@ void SimplicialCholeskyBase<Derived>::factorize_preordered(const CholMatrixType&
     for (typename CholMatrixType::InnerIterator it(ap, k); it; ++it) {
       StorageIndex i = it.index();
       if (i <= k) {
-        y[i] += numext::conj(it.value()); /* scatter A(i,k) into Y (sum duplicates) */
+        y[i] += getSymm(it.value()); /* scatter A(i,k) into Y (sum duplicates) */
         Index len;
         for (len = 0; tags[i] != k; i = m_parent[i]) {
           pattern[len++] = i; /* L(k,i) is nonzero */
@@ -109,8 +109,8 @@ void SimplicialCholeskyBase<Derived>::factorize_preordered(const CholMatrixType&
 
     /* compute numerical values kth row of L (a sparse triangular solve) */
 
-    RealScalar d =
-        numext::real(y[k]) * m_shiftScale + m_shiftOffset;  // get D(k,k), apply the shift function, and clear Y(k)
+    DiagonalScalar d =
+        getDiag(y[k]) * m_shiftScale + m_shiftOffset;  // get D(k,k), apply the shift function, and clear Y(k)
     y[k] = Scalar(0);
     for (; top < size; ++top) {
       Index i = pattern[top]; /* pattern[top:n-1] is pattern of L(:,k) */
@@ -120,14 +120,14 @@ void SimplicialCholeskyBase<Derived>::factorize_preordered(const CholMatrixType&
       /* the nonzero entry L(k,i) */
       Scalar l_ki;
       if (DoLDLT)
-        l_ki = yi / numext::real(m_diag[i]);
+        l_ki = yi / getDiag(m_diag[i]);
       else
         yi = l_ki = yi / Lx[Lp[i]];
 
       Index p2 = Lp[i] + m_nonZerosPerCol[i];
       Index p;
-      for (p = Lp[i] + (DoLDLT ? 0 : 1); p < p2; ++p) y[Li[p]] -= numext::conj(Lx[p]) * yi;
-      d -= numext::real(l_ki * numext::conj(yi));
+      for (p = Lp[i] + (DoLDLT ? 0 : 1); p < p2; ++p) y[Li[p]] -= getSymm(Lx[p]) * yi;
+      d -= getDiag(l_ki * getSymm(yi));
       Li[p] = k; /* store L(k,i) in column form of L */
       Lx[p] = l_ki;
       ++m_nonZerosPerCol[i]; /* increment count of nonzeros in col i */
@@ -141,7 +141,7 @@ void SimplicialCholeskyBase<Derived>::factorize_preordered(const CholMatrixType&
     } else {
       Index p = Lp[k] + m_nonZerosPerCol[k]++;
       Li[p] = k; /* store L(k,k) = sqrt (d) in column k */
-      if (d <= RealScalar(0)) {
+      if (NonHermitian ? d == RealScalar(0) : numext::real(d) <= RealScalar(0)) {
         ok = false; /* failure, matrix is not positive definite */
         break;
       }
diff --git a/Eigen/src/SparseCore/SparseSelfAdjointView.h b/Eigen/src/SparseCore/SparseSelfAdjointView.h
index 2f087c9c4..3402baeb6 100644
--- a/Eigen/src/SparseCore/SparseSelfAdjointView.h
+++ b/Eigen/src/SparseCore/SparseSelfAdjointView.h
@@ -34,13 +34,13 @@ namespace internal {
 template <typename MatrixType, unsigned int Mode>
 struct traits<SparseSelfAdjointView<MatrixType, Mode> > : traits<MatrixType> {};
 
-template <int SrcMode, int DstMode, typename MatrixType, int DestOrder>
+template <int SrcMode, int DstMode, bool NonHermitian, typename MatrixType, int DestOrder>
 void permute_symm_to_symm(
     const MatrixType& mat,
     SparseMatrix<typename MatrixType::Scalar, DestOrder, typename MatrixType::StorageIndex>& _dest,
     const typename MatrixType::StorageIndex* perm = 0);
 
-template <int Mode, typename MatrixType, int DestOrder>
+template <int Mode, bool NonHermitian, typename MatrixType, int DestOrder>
 void permute_symm_to_fullsymm(
     const MatrixType& mat,
     SparseMatrix<typename MatrixType::Scalar, DestOrder, typename MatrixType::StorageIndex>& _dest,
@@ -234,7 +234,7 @@ struct Assignment<DstXprType, SrcXprType, Functor, SparseSelfAdjoint2Sparse> {
   template <typename DestScalar, int StorageOrder>
   static void run(SparseMatrix<DestScalar, StorageOrder, StorageIndex>& dst, const SrcXprType& src,
                   const AssignOpType& /*func*/) {
-    internal::permute_symm_to_fullsymm<SrcXprType::Mode>(src.matrix(), dst);
+    internal::permute_symm_to_fullsymm<SrcXprType::Mode, false>(src.matrix(), dst);
   }
 
   // FIXME: the handling of += and -= in sparse matrices should be cleanup so that next two overloads could be reduced
@@ -405,7 +405,7 @@ struct product_evaluator<Product<Lhs, RhsView, DefaultProduct>, ProductTag, Spar
  ***************************************************************************/
 namespace internal {
 
-template <int Mode, typename MatrixType, int DestOrder>
+template <int Mode, bool NonHermitian, typename MatrixType, int DestOrder>
 void permute_symm_to_fullsymm(
     const MatrixType& mat,
     SparseMatrix<typename MatrixType::Scalar, DestOrder, typename MatrixType::StorageIndex>& _dest,
@@ -476,13 +476,13 @@ void permute_symm_to_fullsymm(
         dest.valuePtr()[k] = it.value();
         k = count[ip]++;
         dest.innerIndexPtr()[k] = jp;
-        dest.valuePtr()[k] = numext::conj(it.value());
+        dest.valuePtr()[k] = (NonHermitian ? it.value() : numext::conj(it.value()));
       }
     }
   }
 }
 
-template <int SrcMode_, int DstMode_, typename MatrixType, int DstOrder>
+template <int SrcMode_, int DstMode_, bool NonHermitian, typename MatrixType, int DstOrder>
 void permute_symm_to_symm(const MatrixType& mat,
                           SparseMatrix<typename MatrixType::Scalar, DstOrder, typename MatrixType::StorageIndex>& _dest,
                           const typename MatrixType::StorageIndex* perm) {
@@ -534,7 +534,7 @@ void permute_symm_to_symm(const MatrixType& mat,
 
       if (!StorageOrderMatch) std::swap(ip, jp);
       if (((int(DstMode) == int(Lower) && ip < jp) || (int(DstMode) == int(Upper) && ip > jp)))
-        dest.valuePtr()[k] = numext::conj(it.value());
+        dest.valuePtr()[k] = (NonHermitian ? it.value() : numext::conj(it.value()));
       else
         dest.valuePtr()[k] = it.value();
     }
@@ -595,14 +595,14 @@ struct Assignment<DstXprType, SparseSymmetricPermutationProduct<MatrixType, Mode
                   const internal::assign_op<Scalar, typename MatrixType::Scalar>&) {
     // internal::permute_symm_to_fullsymm<Mode>(m_matrix,_dest,m_perm.indices().data());
     SparseMatrix<Scalar, (Options & RowMajor) == RowMajor ? ColMajor : RowMajor, DstIndex> tmp;
-    internal::permute_symm_to_fullsymm<Mode>(src.matrix(), tmp, src.perm().indices().data());
+    internal::permute_symm_to_fullsymm<Mode, false>(src.matrix(), tmp, src.perm().indices().data());
     dst = tmp;
   }
 
   template <typename DestType, unsigned int DestMode>
   static void run(SparseSelfAdjointView<DestType, DestMode>& dst, const SrcXprType& src,
                   const internal::assign_op<Scalar, typename MatrixType::Scalar>&) {
-    internal::permute_symm_to_symm<Mode, DestMode>(src.matrix(), dst.matrix(), src.perm().indices().data());
+    internal::permute_symm_to_symm<Mode, DestMode, false>(src.matrix(), dst.matrix(), src.perm().indices().data());
   }
 };
 
diff --git a/test/simplicial_cholesky.cpp b/test/simplicial_cholesky.cpp
index ca6749671..ed9321871 100644
--- a/test/simplicial_cholesky.cpp
+++ b/test/simplicial_cholesky.cpp
@@ -20,6 +20,12 @@ void test_simplicial_cholesky_T() {
   SimplicialLDLT<SparseMatrixType, Upper> ldlt_colmajor_upper_amd;
   SimplicialLDLT<SparseMatrixType, Lower, NaturalOrdering<I_> > ldlt_colmajor_lower_nat;
   SimplicialLDLT<SparseMatrixType, Upper, NaturalOrdering<I_> > ldlt_colmajor_upper_nat;
+  SimplicialNonHermitianLLT<SparseMatrixType, Lower> nhllt_colmajor_lower_amd;
+  SimplicialNonHermitianLLT<SparseMatrixType, Upper> nhllt_colmajor_upper_amd;
+  SimplicialNonHermitianLDLT<SparseMatrixType, Lower> nhldlt_colmajor_lower_amd;
+  SimplicialNonHermitianLDLT<SparseMatrixType, Upper> nhldlt_colmajor_upper_amd;
+  SimplicialNonHermitianLDLT<SparseMatrixType, Lower, NaturalOrdering<I_> > nhldlt_colmajor_lower_nat;
+  SimplicialNonHermitianLDLT<SparseMatrixType, Upper, NaturalOrdering<I_> > nhldlt_colmajor_upper_nat;
 
   check_sparse_spd_solving(chol_colmajor_lower_amd);
   check_sparse_spd_solving(chol_colmajor_upper_amd);
@@ -27,6 +33,10 @@ void test_simplicial_cholesky_T() {
   check_sparse_spd_solving(llt_colmajor_upper_amd);
   check_sparse_spd_solving(ldlt_colmajor_lower_amd);
   check_sparse_spd_solving(ldlt_colmajor_upper_amd);
+  check_sparse_nonhermitian_solving(nhllt_colmajor_lower_amd);
+  check_sparse_nonhermitian_solving(nhllt_colmajor_upper_amd);
+  check_sparse_nonhermitian_solving(nhldlt_colmajor_lower_amd);
+  check_sparse_nonhermitian_solving(nhldlt_colmajor_upper_amd);
 
   check_sparse_spd_determinant(chol_colmajor_lower_amd);
   check_sparse_spd_determinant(chol_colmajor_upper_amd);
@@ -34,9 +44,15 @@ void test_simplicial_cholesky_T() {
   check_sparse_spd_determinant(llt_colmajor_upper_amd);
   check_sparse_spd_determinant(ldlt_colmajor_lower_amd);
   check_sparse_spd_determinant(ldlt_colmajor_upper_amd);
+  check_sparse_nonhermitian_determinant(nhllt_colmajor_lower_amd);
+  check_sparse_nonhermitian_determinant(nhllt_colmajor_upper_amd);
+  check_sparse_nonhermitian_determinant(nhldlt_colmajor_lower_amd);
+  check_sparse_nonhermitian_determinant(nhldlt_colmajor_upper_amd);
 
   check_sparse_spd_solving(ldlt_colmajor_lower_nat, (std::min)(300, EIGEN_TEST_MAX_SIZE), 1000);
   check_sparse_spd_solving(ldlt_colmajor_upper_nat, (std::min)(300, EIGEN_TEST_MAX_SIZE), 1000);
+  check_sparse_nonhermitian_solving(nhldlt_colmajor_lower_nat, (std::min)(300, EIGEN_TEST_MAX_SIZE), 1000);
+  check_sparse_nonhermitian_solving(nhldlt_colmajor_upper_nat, (std::min)(300, EIGEN_TEST_MAX_SIZE), 1000);
 }
 
 EIGEN_DECLARE_TEST(simplicial_cholesky) {
diff --git a/test/sparse_solver.h b/test/sparse_solver.h
index 033df835f..50cb463de 100644
--- a/test/sparse_solver.h
+++ b/test/sparse_solver.h
@@ -484,6 +484,96 @@ void check_sparse_spd_determinant(Solver& solver) {
   }
 }
 
+template <typename Solver, typename DenseMat>
+int generate_sparse_nonhermitian_problem(Solver&, typename Solver::MatrixType& A, typename Solver::MatrixType& halfA,
+                                         DenseMat& dA, int maxSize = 300) {
+  typedef typename Solver::MatrixType Mat;
+  typedef typename Mat::Scalar Scalar;
+  typedef Matrix<Scalar, Dynamic, Dynamic> DenseMatrix;
+
+  int size = internal::random<int>(1, maxSize);
+  double density = (std::max)(8. / static_cast<double>(size * size), 0.01);
+
+  Mat M(size, size);
+  DenseMatrix dM(size, size);
+
+  initSparse<Scalar>(density, dM, M, ForceNonZeroDiag);
+
+  A = M * M.transpose();
+  dA = dM * dM.transpose();
+
+  halfA.resize(size, size);
+  if (Solver::UpLo == (Lower | Upper))
+    halfA = A;
+  else
+    halfA = A.template triangularView<Solver::UpLo>();
+
+  return size;
+}
+
+template <typename Solver>
+void check_sparse_nonhermitian_solving(Solver& solver, int maxSize = (std::min)(300, EIGEN_TEST_MAX_SIZE),
+                                       int maxRealWorldSize = 100000) {
+  typedef typename Solver::MatrixType Mat;
+  typedef typename Mat::Scalar Scalar;
+  typedef typename Mat::StorageIndex StorageIndex;
+  typedef SparseMatrix<Scalar, ColMajor, StorageIndex> SpMat;
+  typedef SparseVector<Scalar, 0, StorageIndex> SpVec;
+  typedef Matrix<Scalar, Dynamic, Dynamic> DenseMatrix;
+  typedef Matrix<Scalar, Dynamic, 1> DenseVector;
+
+  // generate the problem
+  Mat A, halfA;
+  DenseMatrix dA;
+  for (int i = 0; i < g_repeat; i++) {
+    int size = generate_sparse_nonhermitian_problem(solver, A, halfA, dA, maxSize);
+
+    // generate the right hand sides
+    int rhsCols = internal::random<int>(1, 16);
+    double density = (std::max)(8. / static_cast<double>(size * rhsCols), 0.1);
+    SpMat B(size, rhsCols);
+    DenseVector b = DenseVector::Random(size);
+    DenseMatrix dB(size, rhsCols);
+    initSparse<Scalar>(density, dB, B, ForceNonZeroDiag);
+    SpVec c = B.col(0);
+    DenseVector dc = dB.col(0);
+
+    CALL_SUBTEST(check_sparse_solving(solver, A, b, dA, b));
+    CALL_SUBTEST(check_sparse_solving(solver, halfA, b, dA, b));
+    CALL_SUBTEST(check_sparse_solving(solver, A, dB, dA, dB));
+    CALL_SUBTEST(check_sparse_solving(solver, halfA, dB, dA, dB));
+    CALL_SUBTEST(check_sparse_solving(solver, A, B, dA, dB));
+    CALL_SUBTEST(check_sparse_solving(solver, halfA, B, dA, dB));
+    CALL_SUBTEST(check_sparse_solving(solver, A, c, dA, dc));
+    CALL_SUBTEST(check_sparse_solving(solver, halfA, c, dA, dc));
+
+    // check only once
+    if (i == 0) {
+      b = DenseVector::Zero(size);
+      check_sparse_solving(solver, A, b, dA, b);
+    }
+  }
+
+  EIGEN_UNUSED_VARIABLE(maxRealWorldSize);
+}
+
+template <typename Solver>
+void check_sparse_nonhermitian_determinant(Solver& solver) {
+  typedef typename Solver::MatrixType Mat;
+  typedef typename Mat::Scalar Scalar;
+  typedef Matrix<Scalar, Dynamic, Dynamic> DenseMatrix;
+
+  // generate the problem
+  Mat A, halfA;
+  DenseMatrix dA;
+  generate_sparse_nonhermitian_problem(solver, A, halfA, dA, 30);
+
+  for (int i = 0; i < g_repeat; i++) {
+    check_sparse_determinant(solver, A, dA);
+    check_sparse_determinant(solver, halfA, dA);
+  }
+}
+
 template <typename Solver>
 void check_sparse_zero_matrix(Solver& solver) {
   typedef typename Solver::MatrixType Mat;