Introduction to Computational Linear Algebra

Algorithms for the Eigenvalue Problem 115

5.2 QR Method for a Non-Symmetric Matrix

In this section, it is assumed that the matrix A is real and that compu-

tations with complex numbers will be delayed as long as possible. When the

matrix is non-symmetric, some eigenvalues can be complex and therefore at

least the final step of the algorithm must cope with complex arithmetic. This

strategy is preferred to the direct use of complex numbers since the efficiency

of complex arithmetic is lower than real arithmetic

1

. However, it is possible to

consider the complex version of the algorithms by a straight transformation

in which the inner-product u

∗

v of two complex ...