Since Eigen version 3.3 and later, any F77 compatible BLAS or LAPACK libraries can be used as backends for dense matrix products and dense matrix decompositions. For instance, one can use Intel® MKL, Apple's Accelerate framework on OSX, OpenBLAS, Netlib LAPACK, etc.

Do not miss this page for further discussions on the specific use of Intel® MKL (also includes VML, PARDISO, etc.)

In order to use an external BLAS and/or LAPACK library, you must link you own application to the respective libraries and their dependencies. For LAPACK, you must also link to the standard Lapacke library, which is used as a convenient think layer between Eigen's C++ code and LAPACK F77 interface. Then you must activate their usage by defining one or multiple of the following macros (before including any Eigen's header):

Note: For Mac users, in order to use the lapack version shipped with the Accelerate framework, you also need the lapacke library. Using MacPorts, this is as easy as:
sudo port install lapack

and then use the following link flags: -framework Accelerate /opt/local/lib/lapack/liblapacke.dylib

`EIGEN_USE_BLAS`	Enables the use of external BLAS level 2 and 3 routines (compatible with any F77 BLAS interface)
`EIGEN_USE_LAPACKE`	Enables the use of external Lapack routines via the Lapacke C interface to Lapack (compatible with any F77 LAPACK interface)
`EIGEN_USE_LAPACKE_STRICT`	Same as `EIGEN_USE_LAPACKE` but algorithms of lower numerical robustness are disabled. This currently concerns only JacobiSVD which otherwise would be replaced by `gesvd` that is less robust than Jacobi rotations.

When doing so, a number of Eigen's algorithms are silently substituted with calls to BLAS or LAPACK routines. These substitutions apply only for Dynamic or large enough objects with one of the following four standard scalar types: float, double, complex<float>, and complex<double>. Operations on other scalar types or mixing reals and complexes will continue to use the built-in algorithms.

The breadth of Eigen functionality that can be substituted is listed in the table below.

Functional domain	Code example	BLAS/LAPACK routines
Matrix-matrix operations `EIGEN_USE_BLAS`	m1m2.transpose(); m1.selfadjointView<Lower>()m2; m1m2.triangularView<Upper>(); m1.selfadjointView<Lower>().rankUpdate(m2,1.0); m1 Matrix3d m1 Definition:* IOFormat.cpp:2 m2 MatrixType m2(n_dims) Eigen::MatrixBase::selfadjointView SelfAdjointViewReturnType< UpLo >::Type selfadjointView() Eigen::Lower @ Lower Definition: Constants.h:211 Eigen::Upper @ Upper Definition: Constants.h:213	?gemm ?symm/?hemm ?trmm dsyrk/ssyrk ssyrk int BLASFUNC() ssyrk(const char , const char , const int , const int , const float , const float , const int , const float , float , const int ) dsyrk int BLASFUNC() dsyrk(const char , const char , const int , const int , const double , const double , const int , const double , double , const int ) Eigen::internal::gemm void gemm(const DataMapper &res, const Scalar blockA, const Scalar blockB, Index rows, Index depth, Index cols, Scalar alpha, Index strideA, Index strideB, Index offsetA, Index offsetB) Definition: MatrixProduct.h:2250
Matrix-vector operations `EIGEN_USE_BLAS`	m1.adjoint()b; m1.selfadjointView<Lower>()b; m1.triangularView<Upper>()b; b Array< int, 3, 1 > b Definition:* Array_variadic_ctor_cxx11.cpp:2 Eigen::MatrixBase::triangularView TriangularViewReturnType< Mode >::Type triangularView() Eigen::MatrixBase::adjoint const AdjointReturnType adjoint() const Definition: Transpose.h:223	?gemv ?symv/?hemv ?trmv
LU decomposition `EIGEN_USE_LAPACKE` `EIGEN_USE_LAPACKE_STRICT`	v1 = m1.lu().solve(v2); v2 Map< RowVectorXf > v2(M2.data(), M2.size()) v1 M1<< 1, 2, 3, 4, 5, 6, 7, 8, 9;Map< RowVectorXf > v1(M1.data(), M1.size()) Eigen::MatrixBase::lu const PartialPivLU< PlainObject, PermutationIndex > lu() const	?getrf
Cholesky decomposition `EIGEN_USE_LAPACKE` `EIGEN_USE_LAPACKE_STRICT`	v1 = m2.selfadjointView<Upper>().llt().solve(v2);	?potrf
QR decomposition `EIGEN_USE_LAPACKE` `EIGEN_USE_LAPACKE_STRICT`	m1.householderQr(); m1.colPivHouseholderQr(); Eigen::MatrixBase::colPivHouseholderQr const ColPivHouseholderQR< PlainObject, PermutationIndex > colPivHouseholderQr() const Eigen::MatrixBase::householderQr const HouseholderQR< PlainObject > householderQr() const Definition: HouseholderQR.h:527	?geqrf ?geqp3
Singular value decomposition `EIGEN_USE_LAPACKE`	JacobiSVD<MatrixXd, ComputeThinV> svd; svd.compute(m1); svd cout<< "Here is the matrix m:"<< endl<< m<< endl;JacobiSVD< MatrixXf, ComputeThinU\|ComputeThinV > svd(m)	?gesvd
Singular value decomposition `EIGEN_USE_LAPACKE` `EIGEN_USE_LAPACKE_STRICT`	BDCSVD<MatrixXd> svd; svd.compute(m1);	?gesdd
Eigen-value decompositions `EIGEN_USE_LAPACKE` `EIGEN_USE_LAPACKE_STRICT`	EigenSolver<MatrixXd> es(m1); ComplexEigenSolver<MatrixXcd> ces(m1); SelfAdjointEigenSolver<MatrixXd> saes(m1+m1.transpose()); GeneralizedSelfAdjointEigenSolver<MatrixXd> gsaes(m1+m1.transpose(),m2+m2.transpose()); ces cout<< "Here is a random 4x4 matrix, A:"<< endl<< A<< endl<< endl;ComplexEigenSolver< MatrixXcf > ces Definition: ComplexEigenSolver_compute.cpp:4 es EigenSolver< MatrixXf > es Definition: EigenSolver_compute.cpp:1 Eigen::DenseBase::transpose TransposeReturnType transpose() Definition: Transpose.h:184	?gees ?gees ?syev/?heev ?syev/?heev, ?potrf
Schur decomposition `EIGEN_USE_LAPACKE` `EIGEN_USE_LAPACKE_STRICT`	RealSchur<MatrixXd> schurR(m1); ComplexSchur<MatrixXcd> schurC(m1);	?gees

In the examples, m1 and m2 are dense matrices and v1 and v2 are dense vectors.