k], where these values are real or occur in complex conjugate pairs, are the Arnoldi Ritz values at iteration k if and only if the linear system [C.

2] is real and has either two real eigenvalues or a pair of complex conjugate eigenvalues.

2 display an example (Example 1) of a matrix of order 4 with two real eigenvalues on each side of the real part of a pair of complex conjugate eigenvalues.

The matrix A has either two pairs of complex conjugate eigenvalues ([[lambda].

4 displays the feasible region and the boundary for a case with two pairs of complex conjugate eigenvalues and a real eigenvalue inside the field of values (Example 2).

epsilon]](z) has one simple zero in each of the intervals (-[pi]/[sigma], 0), (0, [pi]/[sigma]) and a pair of complex conjugate zeros close to the point 0.

Analogous to the above case of pair of complex conjugate zeros, by Theorem B2 we conclude [[?

epsilon]](z), being real, can have pairs of complex conjugate zeros with the same multiplicities.

epsilon]](z) in this domain including the complex conjugate will be in [[union].

k] is real, where k is the index of a real parameter or of the second element in a complex conjugate pair {[T.

l] is real and the complex shift parameters appear in complex conjugate pairs, then the matrices [Z.

1 is applied to a matrix B that has a complex conjugate pair of eigenvalues, then [Q.

It can be shown that the sum of entries on the diagonal of B which correspond to the complex conjugate pair converge linearly to the sum of the real parts of the pair.

j+1], even when summed, may not reveal any information about the real parts of the eigenvalues of B, and it is tempting to assert that the discrete QR algorithm will fail in the case of complex conjugate eigenvalues.

If the Jacobian matrix A contains a pair of complex conjugate eigenvalues, then its (real) Schur form will be block upper-triangular with 2 x 2 blocks, the eigenvalues of which correspond to each pair of complex eigenvalues.