diff --git a/Eigen/src/Core/Product.h b/Eigen/src/Core/Product.h
index 6bad832e0..37683e3c2 100644
--- a/Eigen/src/Core/Product.h
+++ b/Eigen/src/Core/Product.h
@@ -21,7 +21,7 @@ class ProductImpl;
 namespace internal {
 
 template <typename Lhs, typename Rhs, int Option>
-struct traits<Product<Lhs, Rhs, Option> > {
+struct traits<Product<Lhs, Rhs, Option>> {
   typedef remove_all_t<Lhs> LhsCleaned;
   typedef remove_all_t<Rhs> RhsCleaned;
   typedef traits<LhsCleaned> LhsTraits;
@@ -55,6 +55,129 @@ struct traits<Product<Lhs, Rhs, Option> > {
   };
 };
 
+struct TransposeProductEnum {
+  // convenience enumerations to specialize transposed products
+  enum : int {
+    Default = 0x00,
+    Matrix = 0x01,
+    Permutation = 0x02,
+    MatrixMatrix = (Matrix << 8) | Matrix,
+    MatrixPermutation = (Matrix << 8) | Permutation,
+    PermutationMatrix = (Permutation << 8) | Matrix
+  };
+};
+template <typename Xpr>
+struct TransposeKind {
+  static constexpr int Kind = is_matrix_base_xpr<Xpr>::value        ? TransposeProductEnum::Matrix
+                              : is_permutation_base_xpr<Xpr>::value ? TransposeProductEnum::Permutation
+                                                                    : TransposeProductEnum::Default;
+};
+
+template <typename Lhs, typename Rhs>
+struct TransposeProductKind {
+  static constexpr int Kind = (TransposeKind<Lhs>::Kind << 8) | TransposeKind<Rhs>::Kind;
+};
+
+template <typename Lhs, typename Rhs, int Option, int Kind = TransposeProductKind<Lhs, Rhs>::Kind>
+struct product_transpose_helper {
+  // by default, don't optimize the transposed product
+  using Derived = Product<Lhs, Rhs, Option>;
+  using Scalar = typename Derived::Scalar;
+  using TransposeType = Transpose<const Derived>;
+  using ConjugateTransposeType = CwiseUnaryOp<scalar_conjugate_op<Scalar>, TransposeType>;
+  using AdjointType = std::conditional_t<NumTraits<Scalar>::IsComplex, ConjugateTransposeType, TransposeType>;
+
+  // return (lhs * rhs)^T
+  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TransposeType run_transpose(const Derived& derived) {
+    return TransposeType(derived);
+  }
+  // return (lhs * rhs)^H
+  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE AdjointType run_adjoint(const Derived& derived) {
+    return AdjointType(TransposeType(derived));
+  }
+};
+
+template <typename Lhs, typename Rhs, int Option>
+struct product_transpose_helper<Lhs, Rhs, Option, TransposeProductEnum::MatrixMatrix> {
+  // expand the transposed matrix-matrix product
+  using Derived = Product<Lhs, Rhs, Option>;
+
+  using LhsScalar = typename traits<Lhs>::Scalar;
+  using LhsTransposeType = typename DenseBase<Lhs>::ConstTransposeReturnType;
+  using LhsConjugateTransposeType = CwiseUnaryOp<scalar_conjugate_op<LhsScalar>, LhsTransposeType>;
+  using LhsAdjointType =
+      std::conditional_t<NumTraits<LhsScalar>::IsComplex, LhsConjugateTransposeType, LhsTransposeType>;
+
+  using RhsScalar = typename traits<Rhs>::Scalar;
+  using RhsTransposeType = typename DenseBase<Rhs>::ConstTransposeReturnType;
+  using RhsConjugateTransposeType = CwiseUnaryOp<scalar_conjugate_op<RhsScalar>, RhsTransposeType>;
+  using RhsAdjointType =
+      std::conditional_t<NumTraits<RhsScalar>::IsComplex, RhsConjugateTransposeType, RhsTransposeType>;
+
+  using TransposeType = Product<RhsTransposeType, LhsTransposeType, Option>;
+  using AdjointType = Product<RhsAdjointType, LhsAdjointType, Option>;
+
+  // return rhs^T * lhs^T
+  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TransposeType run_transpose(const Derived& derived) {
+    return TransposeType(RhsTransposeType(derived.rhs()), LhsTransposeType(derived.lhs()));
+  }
+  // return rhs^H * lhs^H
+  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE AdjointType run_adjoint(const Derived& derived) {
+    return AdjointType(RhsAdjointType(RhsTransposeType(derived.rhs())),
+                       LhsAdjointType(LhsTransposeType(derived.lhs())));
+  }
+};
+template <typename Lhs, typename Rhs, int Option>
+struct product_transpose_helper<Lhs, Rhs, Option, TransposeProductEnum::PermutationMatrix> {
+  // expand the transposed permutation-matrix product
+  using Derived = Product<Lhs, Rhs, Option>;
+
+  using LhsInverseType = typename PermutationBase<Lhs>::InverseReturnType;
+
+  using RhsScalar = typename traits<Rhs>::Scalar;
+  using RhsTransposeType = typename DenseBase<Rhs>::ConstTransposeReturnType;
+  using RhsConjugateTransposeType = CwiseUnaryOp<scalar_conjugate_op<RhsScalar>, RhsTransposeType>;
+  using RhsAdjointType =
+      std::conditional_t<NumTraits<RhsScalar>::IsComplex, RhsConjugateTransposeType, RhsTransposeType>;
+
+  using TransposeType = Product<RhsTransposeType, LhsInverseType, Option>;
+  using AdjointType = Product<RhsAdjointType, LhsInverseType, Option>;
+
+  // return rhs^T * lhs^-1
+  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TransposeType run_transpose(const Derived& derived) {
+    return TransposeType(RhsTransposeType(derived.rhs()), LhsInverseType(derived.lhs()));
+  }
+  // return rhs^H * lhs^-1
+  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE AdjointType run_adjoint(const Derived& derived) {
+    return AdjointType(RhsAdjointType(RhsTransposeType(derived.rhs())), LhsInverseType(derived.lhs()));
+  }
+};
+template <typename Lhs, typename Rhs, int Option>
+struct product_transpose_helper<Lhs, Rhs, Option, TransposeProductEnum::MatrixPermutation> {
+  // expand the transposed matrix-permutation product
+  using Derived = Product<Lhs, Rhs, Option>;
+
+  using LhsScalar = typename traits<Lhs>::Scalar;
+  using LhsTransposeType = typename DenseBase<Lhs>::ConstTransposeReturnType;
+  using LhsConjugateTransposeType = CwiseUnaryOp<scalar_conjugate_op<LhsScalar>, LhsTransposeType>;
+  using LhsAdjointType =
+      std::conditional_t<NumTraits<LhsScalar>::IsComplex, LhsConjugateTransposeType, LhsTransposeType>;
+
+  using RhsInverseType = typename PermutationBase<Rhs>::InverseReturnType;
+
+  using TransposeType = Product<RhsInverseType, LhsTransposeType, Option>;
+  using AdjointType = Product<RhsInverseType, LhsAdjointType, Option>;
+
+  // return rhs^-1 * lhs^T
+  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TransposeType run_transpose(const Derived& derived) {
+    return TransposeType(RhsInverseType(derived.rhs()), LhsTransposeType(derived.lhs()));
+  }
+  // return rhs^-1 * lhs^H
+  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE AdjointType run_adjoint(const Derived& derived) {
+    return AdjointType(RhsInverseType(derived.rhs()), LhsAdjointType(LhsTransposeType(derived.lhs())));
+  }
+};
+
 }  // end namespace internal
 
 /** \class Product
@@ -93,6 +216,9 @@ class Product
   typedef internal::remove_all_t<LhsNested> LhsNestedCleaned;
   typedef internal::remove_all_t<RhsNested> RhsNestedCleaned;
 
+  using TransposeReturnType = typename internal::product_transpose_helper<Lhs, Rhs, Option>::TransposeType;
+  using AdjointReturnType = typename internal::product_transpose_helper<Lhs, Rhs, Option>::AdjointType;
+
   EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Product(const Lhs& lhs, const Rhs& rhs) : m_lhs(lhs), m_rhs(rhs) {
     eigen_assert(lhs.cols() == rhs.rows() && "invalid matrix product" &&
                  "if you wanted a coeff-wise or a dot product use the respective explicit functions");
@@ -104,6 +230,13 @@ class Product
   EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const LhsNestedCleaned& lhs() const { return m_lhs; }
   EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const RhsNestedCleaned& rhs() const { return m_rhs; }
 
+  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TransposeReturnType transpose() const {
+    return internal::product_transpose_helper<Lhs, Rhs, Option>::run_transpose(*this);
+  }
+  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE AdjointReturnType adjoint() const {
+    return internal::product_transpose_helper<Lhs, Rhs, Option>::run_adjoint(*this);
+  }
+
  protected:
   LhsNested m_lhs;
   RhsNested m_rhs;
@@ -112,12 +245,12 @@ class Product
 namespace internal {
 
 template <typename Lhs, typename Rhs, int Option, int ProductTag = internal::product_type<Lhs, Rhs>::ret>
-class dense_product_base : public internal::dense_xpr_base<Product<Lhs, Rhs, Option> >::type {};
+class dense_product_base : public internal::dense_xpr_base<Product<Lhs, Rhs, Option>>::type {};
 
 /** Conversion to scalar for inner-products */
 template <typename Lhs, typename Rhs, int Option>
 class dense_product_base<Lhs, Rhs, Option, InnerProduct>
-    : public internal::dense_xpr_base<Product<Lhs, Rhs, Option> >::type {
+    : public internal::dense_xpr_base<Product<Lhs, Rhs, Option>>::type {
   typedef Product<Lhs, Rhs, Option> ProductXpr;
   typedef typename internal::dense_xpr_base<ProductXpr>::type Base;
 
diff --git a/Eigen/src/Core/util/XprHelper.h b/Eigen/src/Core/util/XprHelper.h
index 555faa1cc..a6a7d3fbb 100644
--- a/Eigen/src/Core/util/XprHelper.h
+++ b/Eigen/src/Core/util/XprHelper.h
@@ -928,6 +928,12 @@ template <typename XprType, int BlockRows, int BlockCols, bool InnerPanel>
 struct block_xpr_helper<const Block<XprType, BlockRows, BlockCols, InnerPanel>>
     : block_xpr_helper<Block<XprType, BlockRows, BlockCols, InnerPanel>> {};
 
+template <typename XprType>
+struct is_matrix_base_xpr : std::is_base_of<MatrixBase<remove_all_t<XprType>>, remove_all_t<XprType>> {};
+
+template <typename XprType>
+struct is_permutation_base_xpr : std::is_base_of<PermutationBase<remove_all_t<XprType>>, remove_all_t<XprType>> {};
+
 }  // end namespace internal
 
 /** \class ScalarBinaryOpTraits
diff --git a/test/product_notemporary.cpp b/test/product_notemporary.cpp
index 77a296966..c22ea13cd 100644
--- a/test/product_notemporary.cpp
+++ b/test/product_notemporary.cpp
@@ -65,6 +65,8 @@ void product_notemporary(const MatrixType& m) {
 
   VERIFY_EVALUATION_COUNT(m3 = (m1 * m2.adjoint()), 1);
   VERIFY_EVALUATION_COUNT(m3 = (m1 * m2.adjoint()).transpose(), 1);
+  VERIFY_EVALUATION_COUNT(m3.noalias() = (m1 * m2.adjoint()).transpose(), 0);
+  VERIFY_EVALUATION_COUNT(m3.noalias() = (m1 * m2.transpose()).adjoint(), 0);
   VERIFY_EVALUATION_COUNT(m3.noalias() = m1 * m2.adjoint(), 0);
 
   VERIFY_EVALUATION_COUNT(m3 = s1 * (m1 * m2.transpose()), 1);
@@ -75,6 +77,8 @@ void product_notemporary(const MatrixType& m) {
   VERIFY_EVALUATION_COUNT(m3 = m3 - (m1 * m2.adjoint()), 1);
 
   VERIFY_EVALUATION_COUNT(m3 = m3 + (m1 * m2.adjoint()).transpose(), 1);
+  VERIFY_EVALUATION_COUNT(m3.noalias() = m3 + (m1 * m2.adjoint()).transpose(), 0);
+  VERIFY_EVALUATION_COUNT(m3.noalias() = m3 + (m1 * m2.transpose()).adjoint(), 0);
   VERIFY_EVALUATION_COUNT(m3.noalias() = m3 + m1 * m2.transpose(), 0);
   VERIFY_EVALUATION_COUNT(m3.noalias() += m3 + m1 * m2.transpose(), 0);
   VERIFY_EVALUATION_COUNT(m3.noalias() -= m3 + m1 * m2.transpose(), 0);