diff --git a/src/polysolve/linear/CMakeLists.txt b/src/polysolve/linear/CMakeLists.txt
index a742a357..43d2905e 100644
--- a/src/polysolve/linear/CMakeLists.txt
+++ b/src/polysolve/linear/CMakeLists.txt
@@ -15,6 +15,8 @@ set(SOURCES
     Pardiso.hpp
     SaddlePointSolver.cpp
     SaddlePointSolver.hpp
+    SPQR.cpp
+    SPQR.hpp
 )
 
 source_group(TREE "${CMAKE_CURRENT_SOURCE_DIR}" PREFIX "Source Files" FILES ${SOURCES})
diff --git a/src/polysolve/linear/SPQR.cpp b/src/polysolve/linear/SPQR.cpp
new file mode 100644
index 00000000..f94b123e
--- /dev/null
+++ b/src/polysolve/linear/SPQR.cpp
@@ -0,0 +1,20 @@
+#include "SPQR.hpp"
+
+namespace polysolve::linear
+{
+    template <>
+    void EigenDirect<Eigen::SPQR<StiffnessMatrix>>::analyze_pattern(const StiffnessMatrix &A, const int precond_num)
+    {
+        m_Solver.compute(A);
+    }
+    template <>
+    void EigenDirect<Eigen::SPQR<StiffnessMatrix>>::factorize(const StiffnessMatrix &A)
+    {
+        m_Solver.compute(A);
+        if (m_Solver.info() == Eigen::NumericalIssue)
+        {
+            throw std::runtime_error("[EigenDirect] NumericalIssue encountered.");
+        }
+    }
+
+} // namespace polysolve::linear
diff --git a/src/polysolve/linear/SPQR.hpp b/src/polysolve/linear/SPQR.hpp
new file mode 100644
index 00000000..acedb6c6
--- /dev/null
+++ b/src/polysolve/linear/SPQR.hpp
@@ -0,0 +1,96 @@
+#pragma once
+#ifdef POLYSOLVE_WITH_SPQR
+#include <Eigen/Sparse>
+#include <Eigen/SPQRSupport>
+#include "EigenSolver.hpp"
+#include "Solver.hpp"
+namespace polysolve::linear
+{
+    template <>
+    void EigenDirect<Eigen::SPQR<StiffnessMatrix>>::analyze_pattern(const StiffnessMatrix &A, const int precond_num);
+    template <>
+    void EigenDirect<Eigen::SPQR<StiffnessMatrix>>::factorize(const StiffnessMatrix &A);
+
+    class SPQRSolver : public EigenDirect<Eigen::SPQR<StiffnessMatrix>>
+    {
+
+        StiffnessMatrix matrixQ() const;
+    };
+} // namespace polysolve::linear
+namespace Eigen
+{
+    template <typename SPQRType, typename Derived>
+    struct SPQR_QSparseProduct;
+    namespace internal
+    {
+        template <typename SPQRType, typename Derived>
+        struct traits<SPQR_QSparseProduct<SPQRType, Derived>>
+        {
+            typedef typename Derived::PlainObject ReturnType;
+        };
+    } // namespace internal
+    template <>
+    struct SPQRMatrixQReturnType<SPQR<polysolve::StiffnessMatrix>>
+    {
+
+        using SPQRType = SPQR<polysolve::StiffnessMatrix>;
+        SPQRMatrixQReturnType(const SPQRType &spqr) : m_spqr(spqr) {}
+        template <typename Derived>
+        SPQR_QProduct<SPQRType, Derived> operator*(const MatrixBase<Derived> &other)
+        {
+            return SPQR_QProduct<SPQRType, Derived>(m_spqr, other.derived(), false);
+        }
+        template <typename Derived>
+        SPQR_QSparseProduct<SPQRType, Derived> operator*(const SparseMatrixBase<Derived> &other)
+        {
+            return SPQR_QSparseProduct<SPQRType, Derived>(m_spqr, other.derived(), false);
+        }
+        SPQRMatrixQTransposeReturnType<SPQRType> adjoint() const
+        {
+            return SPQRMatrixQTransposeReturnType<SPQRType>(m_spqr);
+        }
+        // To use for operations with the transpose of Q
+        SPQRMatrixQTransposeReturnType<SPQRType> transpose() const
+        {
+            return SPQRMatrixQTransposeReturnType<SPQRType>(m_spqr);
+        }
+        const SPQRType &m_spqr;
+    };
+
+    template <typename SPQRType, typename Derived>
+    struct SPQR_QSparseProduct : ReturnByValue<SPQR_QSparseProduct<SPQRType, Derived>>
+    {
+        struct SPQRTypeWrap : public SPQRType
+        {
+            using SPQRType::m_H;
+            using SPQRType::m_HPinv;
+            using SPQRType::m_HTau;
+        };
+        typedef typename SPQRType::Scalar Scalar;
+        typedef typename SPQRType::StorageIndex StorageIndex;
+        // Define the constructor to get reference to argument types
+        SPQR_QSparseProduct(const SPQRType &spqr, const Derived &other, bool transpose) : m_spqr(spqr), m_other(other), m_transpose(transpose) {}
+
+        const SPQRTypeWrap &spqr_w() const { return reinterpret_cast<const SPQRTypeWrap &>(m_spqr); }
+
+        inline Index rows() const { return m_transpose ? m_spqr.rows() : m_spqr.cols(); }
+        inline Index cols() const { return m_other.cols(); }
+        // Assign to a vector
+        template <typename ResType>
+        void evalTo(ResType &res) const
+        {
+            cholmod_sparse y_cd;
+            cholmod_sparse *x_cd;
+            int method = m_transpose ? SPQR_QTX : SPQR_QX;
+            cholmod_common *cc = m_spqr.cholmodCommon();
+            y_cd = viewAsCholmod(m_other.const_cast_derived());
+            x_cd = SuiteSparseQR_qmult<Scalar>(method, spqr_w().m_H, spqr_w().m_HTau, spqr_w().m_HPinv, &y_cd, cc);
+            res = viewAsEigen<Scalar, ColMajor, StorageIndex>(*x_cd);
+            cholmod_l_free_sparse(&x_cd, cc);
+        }
+        const SPQRType &m_spqr;
+        const Derived &m_other;
+        bool m_transpose;
+    };
+} // namespace Eigen
+#endif
diff --git a/src/polysolve/linear/Solver.cpp b/src/polysolve/linear/Solver.cpp
index e7c0c259..1083b730 100644
--- a/src/polysolve/linear/Solver.cpp
+++ b/src/polysolve/linear/Solver.cpp
@@ -28,22 +28,7 @@ template <typename _MatrixType, typename _OrderingType = COLAMDOrdering<typename
 #include <Eigen/UmfPackSupport>
 #endif
 #ifdef POLYSOLVE_WITH_SPQR
-#include <Eigen/SPQRSupport>
-namespace polysolve::linear {
-    template <>
-    void EigenDirect<Eigen::SPQR<StiffnessMatrix>>::analyze_pattern(const StiffnessMatrix& A, const int precond_num) {
-        m_Solver.compute(A);
-    }
-    template <>
-    void EigenDirect<Eigen::SPQR<StiffnessMatrix>>::factorize(const StiffnessMatrix &A)
-    {
-        m_Solver.compute(A);
-        if (m_Solver.info() == Eigen::NumericalIssue)
-        {
-            throw std::runtime_error("[EigenDirect] NumericalIssue encountered.");
-        }
-    }
-}
+#include "SPQR.hpp"
 #endif
 #ifdef POLYSOLVE_WITH_SUPERLU
 #include <Eigen/SuperLUSupport>
diff --git a/tests/CMakeLists.txt b/tests/CMakeLists.txt
index 8f7b56ca..193a2103 100644
--- a/tests/CMakeLists.txt
+++ b/tests/CMakeLists.txt
@@ -33,6 +33,10 @@ endif()
 include(polyfem-data)
 target_link_libraries(unit_tests PRIVATE polyfem::data)
 
+if(POLYSOLVE_WITH_SPQR)
+    target_link_libraries(unit_tests PRIVATE SuiteSparse::SPQR)
+endif()
+
 ################################################################################
 # Register tests
 ################################################################################
diff --git a/tests/test_linear_solver.cpp b/tests/test_linear_solver.cpp
index 77a9f169..c6bfd9af 100644
--- a/tests/test_linear_solver.cpp
+++ b/tests/test_linear_solver.cpp
@@ -1,6 +1,7 @@
 //////////////////////////////////////////////////////////////////////////
 #include <polysolve/Types.hpp>
 #include <polysolve/linear/FEMSolver.hpp>
+#include <polysolve/linear/SPQR.hpp>
 
 #include <polysolve/Utils.hpp>
 
@@ -856,3 +857,67 @@ TEST_CASE("cusolverdn_5cubes", "[solver]")
         REQUIRE(err < 1e-8);
     }
 }
+
+TEST_CASE("spqr_sparse_product", "[solver]")
+{
+
+    Eigen::MatrixXd A(4, 4);
+    for (int i = 0; i < 4; i++)
+    {
+        A(i, i) = 1.0;
+    }
+    A(0, 1) = 1.0;
+    A(3, 0) = 1.0;
+    std::unique_ptr<Solver> solver;
+    try
+    {
+        solver = Solver::create("Eigen::SPQR", "");
+    }
+    catch (const std::exception &)
+    {
+        return;
+    }
+
+    using Type = EigenDirect<Eigen::SPQR<polysolve::StiffnessMatrix>>;
+
+    Type *typed_solver = dynamic_cast<Type *>(solver.get());
+    REQUIRE(typed_solver != nullptr);
+    // solver->set_parameters(params);
+
+    for (int i = 0; i < 5; ++i)
+    {
+        A = Eigen::MatrixXd::Random(5, 5);
+        auto As = A.sparseView().eval();
+
+        // do a qr so i have a q
+        Eigen::SPQR<StiffnessMatrix> spqr(As);
+        auto Q = spqr.matrixQ();
+
+        // get a random matrix to multiply against
+        Eigen::MatrixXd B(A.rows(), 5);
+        B.setRandom();
+        Eigen::MatrixXd dense = Q * B;
+
+        // make a sparse version
+        auto Bs = B.sparseView().eval();
+        StiffnessMatrix sparse = Q * Bs;
+
+        // check that the result of the product is the same
+        CHECK((dense - sparse).norm() < 1e-10);
+
+        // try to extract the Q matrix as a dense matrix
+        Eigen::MatrixXd Id = Eigen::MatrixXd::Identity(A.rows(), A.rows());
+        Eigen::MatrixXd denseQ = Q * Id;
+
+        // use the product with B to get a weak equivalence once again
+        Eigen::MatrixXd dense2 = denseQ * B;
+        CHECK((dense2 - dense).norm() < 1e-10);
+
+        // now try using a sparse product
+        StiffnessMatrix I(A.rows(), A.rows());
+        I.setIdentity();
+        StiffnessMatrix myQ = Q * I;
+        StiffnessMatrix sparse2 = myQ * Bs;
+        CHECK((sparse2 - sparse).norm() < 1e-10);
+    }
+}