Hello,
Eigen v3.3 ( windows, 64 bit).
I am trying to evaluate Intel MKL's LU decompostion from Eigen ( FullPivLU ) and I follow with this description :https://eigen.tuxfamily.org/dox/TopicUsingIntelMKL.html
it seems that Eigen::FullPivLU is not substituted with calls to Intel MKL ( getrf ) in that case.

Here is the example of the code I use:
#include <iostream>
#define EIGEN_USE_MKL_ALL
#include <Eigen/Dense>
#include <ctime>
using namespace Eigen;
using namespace std;
int main() {
typedef Matrix<double, 5, 3> Matrix5x3;
typedef Matrix<double, 5, 5> Matrix5x5;
Matrix5x3 m = Matrix5x3::Random();
cout << "Here is the matrix m:" << endl << m << endl;
Eigen::FullPivLU<Matrix5x3> lu(m);
cout << "Here is, up to permutations, its LU decomposition matrix:"
<< endl << lu.matrixLU() << endl;
cout << "Here is the L part:" << endl;

Matrix5x5 l = Matrix5x5::Identity();
l.block<5,3>(0,0).triangularView<StrictlyLower>() = lu.matrixLU();
cout << l << endl;
cout << "Here is the U part:" << endl;

Matrix5x3 u = lu.matrixLU().triangularView<Upper>();
cout << u << endl;
cout << "Let us now reconstruct the original matrix m:" << endl;
cout << lu.permutationP().inverse() * l * u * lu.permutationQ().inverse() << endl;
return 0;
}

to check if Eigen's functions has been substituted by MKL's, I set MKL_VERBOSE=1 environment variables and see no results.
using for example matrix-vector, matrix-matrix, JacobiSVD or Cholesky factorization functions, I see all are OK and Eigen's calls are substituted by MKL:

here is example how I see this by using JacobiSVD:
set MKL_VERBOSE=1
....\Eigen>JacobiSVD.exe
Here is the matrix m:
-0.997497 0.617481
0.127171 0.170019
-0.613392 -0.0402539
MKL_VERBOSE Intel(R) MKL 2017.0 Update 3 Product build 20170413 for Intel(R) 64 architecture Intel(R) Advanced Vector Extensions 2 (Intel(R) AVX2)
enabled processors, Win 1.90GHz cdecl intel_thread NMICDev:0
MKL_VERBOSE SGESVD(S,S,3,2,000000583939D190,3,000000583939F750,000000583939D6D0,3,000000583939D3B0,2,00000058391BF630,-1,0) 5.42ms CNR:OFF Dyn:1 FastM
M:1 TID:0 NThr:2 WDiv:HOST: -nan
MKL_VERBOSE SGESVD(S,S,3,2,000000583939D190,3,000000583939F750,000000583939D6D0,3,000000583939D3B0,2,000000583939F180,10,0) 168.89us CNR:OFF Dyn:1 Fas
tMM:1 TID:0 NThr:2 WDiv:HOST:+0.000
Its singular values are:
1.28435
0.386867
Its left singular vectors are the columns of the thin U matrix:
-0.906518 -0.37201
0.034606 -0.536653
-0.420747 0.757372
Its right singular vectors are the columns of the thin V matrix:
0.908419 -0.41806
-0.41806 -0.908419
Now consider this rhs vector:
1
0
0
A least-squares solution of m*x = rhs is:
MKL_VERBOSE SGEMV(N,2,2,00000058391BF3B8,000000583939D010,2,000000583939F690,1,00000058391BF3B4,000000583939F770,1) 18.02us CNR:OFF Dyn:1 FastMM:1 TID
:0 NThr:2 WDiv:HOST:+0.000
-0.239172
1.16861

Indeed, on PartialPivLU is substituted. LU with full pivoting is inherently very slow, but there are intermediates, like LU with rook pivoting. I don't know if MKL provides anything like that though.