is equivalent to the 2n × 2n real problem A −B u u · =λ B A v v () Note that the 2n × 2n matrix in () is symmetric: AT = A and BT = −B if C is Hermitian. Corresponding to a given eigenvalue λ, the vector −v u () | 482 Chapter 11. Eigensystems is equivalent to the 2n x 2n real problem A B u u B A v A v Note that the 2n x 2n matrix in is symmetric AT A and BT B if C is Hermitian. Corresponding to a given eigenvalue A the vector v u is also an eigenvector as you can verify by writing out the two matrix equations implied by . Thus if A1 A2 . An are the eigenvalues of C then the 2n eigenvalues of the augmented problem are Ai Ai A2 A2 . An An each in other words is repeated twice. The eigenvectors are pairs of the form u iv and i u iv that is they are the same up to an inessential phase. Thus we solve the augmented problem and choose one eigenvalue and eigenvector from each pair. These give the eigenvalues and eigenvectors of the original matrix C. Working with the augmented matrix requires a factor of 2 more storage than the original complex matrix. In principle a complex algorithm is also a factor of 2 more efficient in computer time than is the solution of the augmented problem. CITED REFERENCES AND FURTHER READING Wilkinson . and Reinsch C. 1971 Linear Algebra vol. II of Handbook forAutomatic Computation New York Springer-Verlag . 1 Smith . et al. 1976 Matrix Eigensystem Routines EISPACK Guide 2nd ed. vol. 6 of Lecture Notes in Computer Science New York Springer-Verlag . 2 Reduction of a General Matrix to Hessenberg Form The algorithms for symmetric matrices given in the preceding sections are highly satisfactory in practice. By contrast it is impossible to design equally satisfactory algorithms for the nonsymmetric case. There are two reasons for this. First the eigenvalues of a nonsymmetric matrix can be very sensitive to small changes in the matrix elements. Second the matrix itself can be defective so that there is no complete set of eigenvectors. We emphasize that these difficulties are intrinsic properties of certain nonsymmetric matrices and no numerical procedure can cure them. The best we can hope for are procedures
