template<typename _MatrixType>
class Eigen::RealQZ< _MatrixType >

Performs a real QZ decomposition of a pair of square matrices.

This is defined in the Eigenvalues module.

#include <Eigen/Eigenvalues> 

Template Parameters

_MatrixType

the type of the matrix of which we are computing the real QZ decomposition; this is expected to be an instantiation of the Matrix class template.

Given a real square matrices A and B, this class computes the real QZ decomposition: \( A = Q S Z \), \( B = Q T Z \) where Q and Z are real orthogonal matrixes, T is upper-triangular matrix, and S is upper quasi-triangular matrix. An orthogonal matrix is a matrix whose inverse is equal to its transpose, \( U^{-1} = U^T \). A quasi-triangular matrix is a block-triangular matrix whose diagonal consists of 1-by-1 blocks and 2-by-2 blocks where further reduction is impossible due to complex eigenvalues.

The eigenvalues of the pencil \( A - z B \) can be obtained from 1x1 and 2x2 blocks on the diagonals of S and T.

Call the function compute() to compute the real QZ decomposition of a given pair of matrices. Alternatively, you can use the RealQZ(const MatrixType& B, const MatrixType& B, bool computeQZ) constructor which computes the real QZ decomposition at construction time. Once the decomposition is computed, you can use the matrixS(), matrixT(), matrixQ() and matrixZ() functions to retrieve the matrices S, T, Q and Z in the decomposition. If computeQZ==false, some time is saved by not computing matrices Q and Z.

Example:

MatrixXf A = MatrixXf::Random(4,4);
MatrixXf B = MatrixXf::Random(4,4);
RealQZ<MatrixXf> qz(4); // preallocate space for 4x4 matrices
qz.compute(A,B);  // A = Q S Z,  B = Q T Z
 
// print original matrices and result of decomposition
cout << "A:\n" << A << "\n" << "B:\n" << B << "\n";
cout << "S:\n" << qz.matrixS() << "\n" << "T:\n" << qz.matrixT() << "\n";
cout << "Q:\n" << qz.matrixQ() << "\n" << "Z:\n" << qz.matrixZ() << "\n";
 
// verify precision
cout << "\nErrors:"
  << "\n|A-QSZ|: " << (A-qz.matrixQ()*qz.matrixS()*qz.matrixZ()).norm()
  << ", |B-QTZ|: " << (B-qz.matrixQ()*qz.matrixT()*qz.matrixZ()).norm()
  << "\n|QQ* - I|: " << (qz.matrixQ()*qz.matrixQ().adjoint() - MatrixXf::Identity(4,4)).norm()
  << ", |ZZ* - I|: " << (qz.matrixZ()*qz.matrixZ().adjoint() - MatrixXf::Identity(4,4)).norm()
  << "\n";

Output:

A:
   0.68   0.823  -0.444   -0.27
 -0.211  -0.605   0.108  0.0268
  0.566   -0.33 -0.0452   0.904
  0.597   0.536   0.258   0.832
B:
 0.271 -0.967 -0.687  0.998
 0.435 -0.514 -0.198 -0.563
-0.717 -0.726  -0.74 0.0259
 0.214  0.608 -0.782  0.678
S:
-0.668   1.26 -0.598 0.0941
 0.317  -0.27 -0.279   0.64
     0      0 -0.398 -0.164
     0      0      0  -1.12
T:
 -1.55      0  0.342  -0.54
     0   1.01 -0.457  0.128
     0      0  -1.25  0.438
     0      0      0  0.746
Q:
-0.587 -0.138  0.552  0.576
  0.19 -0.208  0.761 -0.585
-0.292  0.918  0.152 -0.223
-0.731  -0.31 -0.306 -0.526
Z:
-0.0204   0.213   -0.78   0.588
 -0.961  -0.184   -0.14  -0.153
 -0.269   0.783   0.462    0.32
-0.0674  -0.555   0.398   0.727
 
Errors:
|A-QSZ|: 1.36e-06, |B-QTZ|: 1.83e-06
|QQ* - I|: 8.18e-07, |ZZ* - I|: 7.36e-07

Note

The implementation is based on the algorithm in "Matrix Computations" by Gene H. Golub and Charles F. Van Loan, and a paper "An algorithm for generalized eigenvalue problems" by C.B.Moler and G.W.Stewart.

See also

class RealSchur, class ComplexSchur, class EigenSolver, class ComplexEigenSolver