    // @(#)root/matrix:$Name:  $:$Id: TMatrixDEigen.cxx,v 1.10 2005/02/15 16:17:09 brun Exp $
    // Authors: Fons Rademakers, Eddy Offermann  Dec 2003

    /*************************************************************************
     * Copyright (C) 1995-2000, Rene Brun and Fons Rademakers.               *
     * All rights reserved.                                                  *
     *                                                                       *
     * For the licensing terms see $ROOTSYS/LICENSE.                         *
     * For the list of contributors see $ROOTSYS/README/CREDITS.             *
     *************************************************************************/

    //////////////////////////////////////////////////////////////////////////
    //                                                                      //
    // TMatrixDEigen                                                        //
    //                                                                      //
    // Eigenvalues and eigenvectors of a real matrix.                       //
    //                                                                      //
    // If A is not symmetric, then the eigenvalue matrix D is block         //
    // diagonal with the real eigenvalues in 1-by-1 blocks and any complex  //
    // eigenvalues, a + i*b, in 2-by-2 blocks, [a, b; -b, a].  That is, if  //
    // the complex eigenvalues look like                                    //
    //                                                                      //
    //     u + iv     .        .          .      .    .                     //
    //       .      u - iv     .          .      .    .                     //
    //       .        .      a + ib       .      .    .                     //
    //       .        .        .        a - ib   .    .                     //
    //       .        .        .          .      x    .                     //
    //       .        .        .          .      .    y                     //
    //                                                                      //
    // then D looks like                                                    //
    //                                                                      //
    //       u        v        .          .      .    .                     //
    //      -v        u        .          .      .    .                     //
    //       .        .        a          b      .    .                     //
    //       .        .       -b          a      .    .                     //
    //       .        .        .          .      x    .                     //
    //       .        .        .          .      .    y                     //
    //                                                                      //
    // This keeps V a real matrix in both symmetric and non-symmetric       //
    // cases, and A*V = V*D.                                                //
    //                                                                      //
    //////////////////////////////////////////////////////////////////////////

    #include "TMatrixDEigen.h"

    ClassImp(TMatrixDEigen)

    //______________________________________________________________________________
     TMatrixDEigen::TMatrixDEigen(const TMatrixD &a)
    {
      Assert(a.IsValid());

      const Int_t nRows  = a.GetNrows();
      const Int_t nCols  = a.GetNcols();
      const Int_t rowLwb = a.GetRowLwb();
      const Int_t colLwb = a.GetColLwb();

      if (nRows != nCols || rowLwb != colLwb)
      {
        Error("TMatrixDEigen(TMatrixD &)","matrix should be square");
        return;
      }

      const Int_t rowUpb = rowLwb+nRows-1;
      fEigenVectors.ResizeTo(rowLwb,rowUpb,rowLwb,rowUpb);
      fEigenValuesRe.ResizeTo(rowLwb,rowUpb);
      fEigenValuesIm.ResizeTo(rowLwb,rowUpb);

      TVectorD ortho;
      Double_t work[kWorkMax];
      if (nRows > kWorkMax) ortho.ResizeTo(nRows);
      else                  ortho.Use(nRows,work);

      TMatrixD H = a;

      // Reduce to Hessenberg form.
      MakeHessenBerg(fEigenVectors,ortho,H);

      // Reduce Hessenberg to real Schur form.
      MakeSchurr(fEigenVectors,fEigenValuesRe,fEigenValuesIm,H);

      // Sort eigenvalues and corresponding vectors in descending order of Re^2+Im^2
      // of the complex eigenvalues .
      Sort(fEigenVectors,fEigenValuesRe,fEigenValuesIm);
    }

    //______________________________________________________________________________
     TMatrixDEigen::TMatrixDEigen(const TMatrixDEigen &another)
    {
      *this = another;
    }

    //______________________________________________________________________________
     void TMatrixDEigen::MakeHessenBerg(TMatrixD &v,TVectorD &ortho,TMatrixD &H)
    {
    // Nonsymmetric reduction to Hessenberg form.
    // This is derived from the Algol procedures orthes and ortran, by Martin and Wilkinson,
    // Handbook for Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
    // Fortran subroutines in EISPACK.

      Double_t *pV = v.GetMatrixArray();
      Double_t *pO = ortho.GetMatrixArray();
      Double_t *pH = H.GetMatrixArray();

      const Int_t n = v.GetNrows();

      const Int_t low  = 0;
      const Int_t high = n-1;

      Int_t i,j,m;
      for (m = low+1; m <= high-1; m++) {
        const Int_t off_m = m*n;

        // Scale column.

        Double_t scale = 0.0;
        for (i = m; i <= high; i++) {
          const Int_t off_i = i*n;
          scale = scale + TMath::Abs(pH[off_i+m-1]);
        }
        if (scale != 0.0) {

          // Compute Householder transformation.

          Double_t h = 0.0;
          for (i = high; i >= m; i--) {
            const Int_t off_i = i*n;
            pO[i] = pH[off_i+m-1]/scale;
            h += pO[i]*pO[i];
          }
          Double_t g = TMath::Sqrt(h);
          if (pO[m] > 0)
            g = -g;
          h = h-pO[m]*g;
          pO[m] = pO[m]-g;

          // Apply Householder similarity transformation
          // H = (I-u*u'/h)*H*(I-u*u')/h)

          for (j = m; j < n; j++) {
            Double_t f = 0.0;
            for (i = high; i >= m; i--) {
              const Int_t off_i = i*n;
              f += pO[i]*pH[off_i+j];
            }
            f = f/h;
            for (i = m; i <= high; i++) {
              const Int_t off_i = i*n;
              pH[off_i+j] -= f*pO[i];
            }
          }

          for (i = 0; i <= high; i++) {
            const Int_t off_i = i*n;
            Double_t f = 0.0;
            for (j = high; j >= m; j--) {
              const Int_t off_i = i*n;
              f += pO[j]*pH[off_i+j];
            }
            f = f/h;
            for (j = m; j <= high; j++)
              pH[off_i+j] -= f*pO[j];
          }
          pO[m] = scale*pO[m];
          pH[off_m+m-1] = scale*g;
        }
      }

      // Accumulate transformations (Algol's ortran).

      for (i = 0; i < n; i++) {
        const Int_t off_i = i*n;
        for (j = 0; j < n; j++)
          pV[off_i+j] = (i == j ? 1.0 : 0.0);
      }

      for (m = high-1; m >= low+1; m--) {
        const Int_t off_m = m*n;
        if (pH[off_m+m-1] != 0.0) {
          for (i = m+1; i <= high; i++) {
            const Int_t off_i = i*n;
            pO[i] = pH[off_i+m-1];
          }
          for (j = m; j <= high; j++) {
            Double_t g = 0.0;
            for (i = m; i <= high; i++) {
              const Int_t off_i = i*n;
              g += pO[i]*pV[off_i+j];
            }
            // Double division avoids possible underflow
            g = (g/pO[m])/pH[off_m+m-1];
            for (i = m; i <= high; i++) {
              const Int_t off_i = i*n;
              pV[off_i+j] += g*pO[i];
            }
          }
        }
      }
    }

    //______________________________________________________________________________
    static Double_t cdivr, cdivi;
    static void cdiv(Double_t xr,Double_t xi,Double_t yr,Double_t yi) {
    // Complex scalar division.
      Double_t r,d;
      if (TMath::Abs(yr) > TMath::Abs(yi)) {
        r = yi/yr;
        d = yr+r*yi;
        cdivr = (xr+r*xi)/d;
        cdivi = (xi-r*xr)/d;
      } else {
        r = yr/yi;
        d = yi+r*yr;
        cdivr = (r*xr+xi)/d;
        cdivi = (r*xi-xr)/d;
      }
    }

    //______________________________________________________________________________
     void TMatrixDEigen::MakeSchurr(TMatrixD &v,TVectorD &d,TVectorD &e,TMatrixD &H)
    {
    // Nonsymmetric reduction from Hessenberg to real Schur form.
    // This is derived from the Algol procedure hqr2, by Martin and Wilkinson,
    // Handbook for Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
    // Fortran subroutine in EISPACK.

      // Initialize

      const Int_t nn = v.GetNrows();
            Int_t n = nn-1;
      const Int_t low = 0;
      const Int_t high = nn-1;
      const Double_t eps = TMath::Power(2.0,-52.0);
      Double_t exshift = 0.0;
      Double_t p=0,q=0,r=0,s=0,z=0,t,w,x,y;

      Double_t *pV = v.GetMatrixArray();
      Double_t *pD = d.GetMatrixArray();
      Double_t *pE = e.GetMatrixArray();
      Double_t *pH = H.GetMatrixArray();

      // Store roots isolated by balanc and compute matrix norm

      Double_t norm = 0.0;
      Int_t i,j,k;
      for (i = 0; i < nn; i++) {
        const Int_t off_i = i*nn;
        if ((i < low) || (i > high)) {
          pD[i] = pH[off_i+i];
          pE[i] = 0.0;
        }
        for (j = TMath::Max(i-1,0); j < nn; j++)
          norm += TMath::Abs(pH[off_i+j]);
      }

      // Outer loop over eigenvalue index

      Int_t iter = 0;
      while (n >= low) {
        const Int_t off_n  = n*nn;
        const Int_t off_n1 = (n-1)*nn;

        // Look for single small sub-diagonal element

        Int_t l = n;
        while (l > low) {
          const Int_t off_l1 = (l-1)*nn;
          const Int_t off_l  = l*nn;
          s = TMath::Abs(pH[off_l1+l-1])+TMath::Abs(pH[off_l+l]);
          if (s == 0.0)
            s = norm;
          if (TMath::Abs(pH[off_l+l-1]) < eps*s)
            break;
          l--;
        }

        // Check for convergence
        // One root found

        if (l == n) {
          pH[off_n+n] = pH[off_n+n]+exshift;
          pD[n] = pH[off_n+n];
          pE[n] = 0.0;
          n--;
          iter = 0;

          // Two roots found

        } else if (l == n-1) {
          w = pH[off_n+n-1]*pH[off_n1+n];
          p = (pH[off_n1+n-1]-pH[off_n+n])/2.0;
          q = p*p+w;
          z = TMath::Sqrt(TMath::Abs(q));
          pH[off_n+n] = pH[off_n+n]+exshift;
          pH[off_n1+n-1] = pH[off_n1+n-1]+exshift;
          x = pH[off_n+n];

          // Double_t pair

          if (q >= 0) {
            if (p >= 0)
              z = p+z;
            else
              z = p-z;
            pD[n-1] = x+z;
            pD[n] = pD[n-1];
            if (z != 0.0)
              pD[n] = x-w/z;
            pE[n-1] = 0.0;
            pE[n] = 0.0;
            x = pH[off_n+n-1];
            s = TMath::Abs(x)+TMath::Abs(z);
            p = x/s;
            q = z/s;
            r = TMath::Sqrt((p*p)+(q*q));
            p = p/r;
            q = q/r;

            // Row modification

            for (j = n-1; j < nn; j++) {
              z = pH[off_n1+j];
              pH[off_n1+j] = q*z+p*pH[off_n+j];
              pH[off_n+j]  = q*pH[off_n+j]-p*z;
            }

            // Column modification

            for (i = 0; i <= n; i++) {
              const Int_t off_i = i*nn;
              z = pH[off_i+n-1];
              pH[off_i+n-1] = q*z+p*pH[off_i+n];
              pH[off_i+n]  = q*pH[off_i+n]-p*z;
            }

            // Accumulate transformations

            for (i = low; i <= high; i++) {
              const Int_t off_i = i*nn;
              z = pV[off_i+n-1];
              pV[off_i+n-1] = q*z+p*pV[off_i+n];
              pV[off_i+n]   = q*pV[off_i+n]-p*z;
            }

            // Complex pair

          } else {
            pD[n-1] = x+p;
            pD[n] = x+p;
            pE[n-1] = z;
            pE[n] = -z;
          }
          n = n-2;
          iter = 0;

          // No convergence yet

        } else {

          // Form shift

          x = pH[off_n+n];
          y = 0.0;
          w = 0.0;
          if (l < n) {
            y = pH[off_n1+n-1];
            w = pH[off_n+n-1]*pH[off_n1+n];
          }

          // Wilkinson's original ad hoc shift

          if (iter == 10) {
            exshift += x;
            for (i = low; i <= n; i++) {
              const Int_t off_i = i*nn;
              pH[off_i+i] -= x;
            }
            s = TMath::Abs(pH[off_n+n-1])+TMath::Abs(pH[off_n1+n-2]);
            x = y = 0.75*s;
            w = -0.4375*s*s;
          }

          // MATLAB's new ad hoc shift

          if (iter == 30) {
             s = (y-x)/2.0;
             s = s*s+w;
             if (s > 0) {
               s = TMath::Sqrt(s);
               if (y<x)
                 s = -s;
               s = x-w/((y-x)/2.0+s);
               for (i = low; i <= n; i++) {
                 const Int_t off_i = i*nn;
                 pH[off_i+i] -= s;
               }
               exshift += s;
               x = y = w = 0.964;
             }
          }

          if (iter++ == 50) {  // (check iteration count here.)
            Error("MakeSchurr","too many iterations");
            break;
          }

          // Look for two consecutive small sub-diagonal elements

          Int_t m = n-2;
          while (m >= l) {
            const Int_t off_m   = m*nn;
            const Int_t off_m_1 = (m-1)*nn;
            const Int_t off_m1  = (m+1)*nn;
            const Int_t off_m2  = (m+2)*nn;
            z = pH[off_m+m];
            r = x-z;
            s = y-z;
            p = (r*s-w)/pH[off_m1+m]+pH[off_m+m+1];
            q = pH[off_m1+m+1]-z-r-s;
            r = pH[off_m2+m+1];
            s = TMath::Abs(p)+TMath::Abs(q)+TMath::Abs(r);
            p = p /s;
            q = q/s;
            r = r/s;
            if (m == l)
              break;
            if (TMath::Abs(pH[off_m+m-1])*(TMath::Abs(q)+TMath::Abs(r)) <
              eps*(TMath::Abs(p)*(TMath::Abs(pH[off_m_1+m-1])+TMath::Abs(z)+
              TMath::Abs(pH[off_m1+m+1]))))
                break;
            m--;
          }

          for (i = m+2; i <= n; i++) {
            const Int_t off_i = i*nn;
            pH[off_i+i-2] = 0.0;
            if (i > m+2)
              pH[off_i+i-3] = 0.0;
          }

          // Double QR step involving rows l:n and columns m:n

          for (k = m; k <= n-1; k++) {
            const Int_t off_k  = k*nn;
            const Int_t off_k1 = (k+1)*nn;
            const Int_t off_k2 = (k+2)*nn;
            const Int_t notlast = (k != n-1);
            if (k != m) {
              p = pH[off_k+k-1];
              q = pH[off_k1+k-1];
              r = (notlast ? pH[off_k2+k-1] : 0.0);
              x = TMath::Abs(p)+TMath::Abs(q)+TMath::Abs(r);
              if (x != 0.0) {
                p = p/x;
                q = q/x;
                r = r/x;
              }
           }
           if (x == 0.0)
             break;
           s = TMath::Sqrt(p*p+q*q+r*r);
           if (p < 0) {
             s = -s;
           }
           if (s != 0) {
             if (k != m)
               pH[off_k+k-1] = -s*x;
              else if (l != m)
                pH[off_k+k-1] = -pH[off_k+k-1];
              p = p+s;
              x = p/s;
              y = q/s;
              z = r/s;
              q = q/p;
              r = r/p;

              // Row modification

              for (j = k; j < nn; j++) {
                p = pH[off_k+j]+q*pH[off_k1+j];
                if (notlast) {
                  p = p+r*pH[off_k2+j];
                  pH[off_k2+j] = pH[off_k2+j]-p*z;
                }
                pH[off_k+j]  = pH[off_k+j]-p*x;
                pH[off_k1+j] = pH[off_k1+j]-p*y;
              }

              // Column modification

              for (i = 0; i <= TMath::Min(n,k+3); i++) {
                const Int_t off_i = i*nn;
                p = x*pH[off_i+k]+y*pH[off_i+k+1];
                if (notlast) {
                  p = p+z*pH[off_i+k+2];
                  pH[off_i+k+2] = pH[off_i+k+2]-p*r;
                }
                pH[off_i+k]   = pH[off_i+k]-p;
                pH[off_i+k+1] = pH[off_i+k+1]-p*q;
              }

              // Accumulate transformations

              for (i = low; i <= high; i++) {
                const Int_t off_i = i*nn;
                p = x*pV[off_i+k]+y*pV[off_i+k+1];
                if (notlast) {
                  p = p+z*pV[off_i+k+2];
                  pV[off_i+k+2] = pV[off_i+k+2]-p*r;
                }
                pV[off_i+k]   = pV[off_i+k]-p;
                pV[off_i+k+1] = pV[off_i+k+1]-p*q;
              }
            }  // (s != 0)
          }  // k loop
        }  // check convergence
      }  // while (n >= low)

      // Backsubstitute to find vectors of upper triangular form

      if (norm == 0.0)
        return;

      for (n = nn-1; n >= 0; n--) {
        p = pD[n];
        q = pE[n];

        // Double_t vector

        const Int_t off_n = n*nn;
        if (q == 0) {
          Int_t l = n;
          pH[off_n+n] = 1.0;
          for (i = n-1; i >= 0; i--) {
            const Int_t off_i  = i*nn;
            const Int_t off_i1 = (i+1)*nn;
            w = pH[off_i+i]-p;
            r = 0.0;
            for (j = l; j <= n; j++) {
              const Int_t off_j = j*nn;
              r = r+pH[off_i+j]*pH[off_j+n];
            }
            if (pE[i] < 0.0) {
              z = w;
              s = r;
            } else {
              l = i;
              if (pE[i] == 0.0) {
                if (w != 0.0)
                  pH[off_i+n] = -r/w;
                else
                  pH[off_i+n] = -r/(eps*norm);

                // Solve real equations

                } else {
                  x = pH[off_i+i+1];
                  y = pH[off_i1+i];
                  q = (pD[i]-p)*(pD[i]-p)+pE[i]*pE[i];
                  t = (x*s-z*r)/q;
                  pH[off_i+n] = t;
                  if (TMath::Abs(x) > TMath::Abs(z))
                    pH[i+1+n] = (-r-w*t)/x;
                  else
                    pH[i+1+n] = (-s-y*t)/z;
                }

                // Overflow control

                t = TMath::Abs(pH[off_i+n]);
                if ((eps*t)*t > 1) {
                  for (j = i; j <= n; j++) {
                    const Int_t off_j = j*nn;
                    pH[off_j+n] = pH[off_j+n]/t;
                  }
                }
             }
          }

          // Complex vector

        } else if (q < 0) {
          Int_t l = n-1;
          const Int_t off_n1 = (n-1)*nn;

          // Last vector component imaginary so matrix is triangular

          if (TMath::Abs(pH[off_n+n-1]) > TMath::Abs(pH[off_n1+n])) {
            pH[off_n1+n-1] = q/pH[off_n+n-1];
            pH[off_n1+n]   = -(pH[off_n+n]-p)/pH[off_n+n-1];
          } else {
            cdiv(0.0,-pH[off_n1+n],pH[off_n1+n-1]-p,q);
            pH[off_n1+n-1] = cdivr;
            pH[off_n1+n]   = cdivi;
          }
          pH[off_n+n-1] = 0.0;
          pH[off_n+n]   = 1.0;
          for (i = n-2; i >= 0; i--) {
            const Int_t off_i  = i*nn;
            const Int_t off_i1 = (i+1)*nn;
            Double_t ra = 0.0;
            Double_t sa = 0.0;
            for (j = l; j <= n; j++) {
              const Int_t off_j = j*nn;
              ra += pH[off_i+j]*pH[off_j+n-1];
              sa += pH[off_i+j]*pH[off_j+n];
            }
            w = pH[off_i+i]-p;

            if (pE[i] < 0.0) {
              z = w;
              r = ra;
              s = sa;
            } else {
              l = i;
              if (pE[i] == 0) {
                cdiv(-ra,-sa,w,q);
                pH[off_i+n-1] = cdivr;
                pH[off_i+n]   = cdivi;
              } else {

                // Solve complex equations

                x = pH[off_i+i+1];
                y = pH[off_i1+i];
                Double_t vr = (pD[i]-p)*(pD[i]-p)+pE[i]*pE[i]-q*q;
                Double_t vi = (pD[i]-p)*2.0*q;
                if ((vr == 0.0) && (vi == 0.0)) {
                  vr = eps*norm*(TMath::Abs(w)+TMath::Abs(q)+
                       TMath::Abs(x)+TMath::Abs(y)+TMath::Abs(z));
                }
                cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
                pH[off_i+n-1] = cdivr;
                pH[off_i+n]   = cdivi;
                if (TMath::Abs(x) > (TMath::Abs(z)+TMath::Abs(q))) {
                  pH[off_i1+n-1] = (-ra-w*pH[off_i+n-1]+q*pH[off_i+n])/x;
                  pH[off_i1+n]   = (-sa-w*pH[off_i+n]-q*pH[off_i+n-1])/x;
                } else {
                  cdiv(-r-y*pH[off_i+n-1],-s-y*pH[off_i+n],z,q);
                  pH[off_i1+n-1] = cdivr;
                  pH[off_i1+n]   = cdivi;
                }
              }

              // Overflow control

              t = TMath::Max(TMath::Abs(pH[off_i+n-1]),TMath::Abs(pH[off_i+n]));
              if ((eps*t)*t > 1) {
                for (j = i; j <= n; j++) {
                  const Int_t off_j = j*nn;
                  pH[off_j+n-1] = pH[off_j+n-1]/t;
                  pH[off_j+n]   = pH[off_j+n]/t;
                }
              }
            }
          }
        }
      }

      // Vectors of isolated roots

      for (i = 0; i < nn; i++) {
        if (i < low || i > high) {
          const Int_t off_i = i*nn;
          for (Int_t j = i; j < nn; j++)
            pV[off_i+j] = pH[off_i+j];
        }
      }

      // Back transformation to get eigenvectors of original matrix

      for (j = nn-1; j >= low; j--) {
        for (i = low; i <= high; i++) {
          const Int_t off_i = i*nn;
          z = 0.0;
          for (k = low; k <= TMath::Min(j,high); k++) {
            const Int_t off_k = k*nn;
            z = z+pV[off_i+k]*pH[off_k+j];
          }
          pV[off_i+j] = z;
        }
      }

    }

    //______________________________________________________________________________
     void TMatrixDEigen::Sort(TMatrixD &v,TVectorD &d,TVectorD &e)
    {
    // Sort eigenvalues and corresponding vectors in descending order of Re^2+Im^2
    // of the complex eigenvalues .

      // Sort eigenvalues and corresponding vectors.
      Double_t *pV = v.GetMatrixArray();
      Double_t *pD = d.GetMatrixArray();
      Double_t *pE = e.GetMatrixArray();

      const Int_t n = v.GetNrows();

      for (Int_t i = 0; i < n-1; i++) {
        Int_t k = i;
        Double_t norm = pD[i]*pD[i]+pE[i]*pE[i];
        Int_t j;
        for (j = i+1; j < n; j++) {
          const Double_t norm_new = pD[j]*pD[j]+pE[j]*pE[j];
          if (norm_new > norm) {
            k = j;
            norm = norm_new;
          }
        }
        if (k != i) {
          Double_t tmp;
          tmp   = pD[k];
          pD[k] = pD[i];
          pD[i] = tmp;
          tmp   = pE[k];
          pE[k] = pE[i];
          pE[i] = tmp;
          for (j = 0; j < n; j++) {
            const Int_t off_j = j*n;
            tmp = pV[off_j+i];
            pV[off_j+i] = pV[off_j+k];
            pV[off_j+k] = tmp;
          }
        }
      }
    }

    //______________________________________________________________________________
    TMatrixDEigen &TMatrixDEigen::operator=(const TMatrixDEigen &source)
    {
      if (this != &source) {
        fEigenVectors.ResizeTo(source.fEigenVectors);
        fEigenValuesRe.ResizeTo(source.fEigenValuesRe);
        fEigenValuesIm.ResizeTo(source.fEigenValuesIm);
      }
      return *this;
    }

    //______________________________________________________________________________
     const TMatrixD TMatrixDEigen::GetEigenValues() const
    {
    // Computes the block diagonal eigenvalue matrix.
    // If the original matrix A is not symmetric, then the eigenvalue
    // matrix D is block diagonal with the real eigenvalues in 1-by-1
    // blocks and any complex eigenvalues,
    //    a + i*b, in 2-by-2 blocks, [a, b; -b, a].
    //  That is, if the complex eigenvalues look like
    //
    //     u + iv     .        .          .      .    .
    //       .      u - iv     .          .      .    .
    //       .        .      a + ib       .      .    .
    //       .        .        .        a - ib   .    .
    //       .        .        .          .      x    .
    //       .        .        .          .      .    y
    //
    // then D looks like
    //
    //     u        v        .          .      .    .
    //    -v        u        .          .      .    .
    //     .        .        a          b      .    .
    //     .        .       -b          a      .    .
    //     .        .        .          .      x    .
    //     .        .        .          .      .    y
    //
    // This keeps V a real matrix in both symmetric and non-symmetric
    // cases, and A*V = V*D.
    //
    // Indexing:
    //  If matrix A has the index/shape (rowLwb,rowUpb,rowLwb,rowUpb)
    //  each eigen-vector must have the shape (rowLwb,rowUpb) .
    //  For convinience, the column index of the eigen-vector matrix
    //  also runs from rowLwb to rowUpb so that the returned matrix
    //  has also index/shape (rowLwb,rowUpb,rowLwb,rowUpb) .
    //
      const Int_t nrows  = fEigenVectors.GetNrows();
      const Int_t rowLwb = fEigenVectors.GetRowLwb();
      const Int_t rowUpb = rowLwb+nrows-1;

      TMatrixD D(rowLwb,rowUpb,rowLwb,rowUpb);

      Double_t *pD = D.GetMatrixArray();
      const Double_t * const pd = fEigenValuesRe.GetMatrixArray();
      const Double_t * const pe = fEigenValuesIm.GetMatrixArray();

      for (Int_t i = 0; i < nrows; i++) {
        const Int_t off_i = i*nrows;
        for (Int_t j = 0; j < nrows; j++)
          pD[off_i+j] = 0.0;
        pD[off_i+i] = pd[i];
        if (pe[i] > 0) {
          pD[off_i+i+1] = pe[i];
        } else if (pe[i] < 0) {
          pD[off_i+i-1] = pe[i];
        }
      }

      return D;
    }