On real eigenvalues of complex matrices
David H Carlson · Pacific Journal of Mathematics · 1965
This paper contains many inter-related results dealing with the genera!question of determination of real eigenvalues of complex matrices.We first discuss the relationship between the number of elementary divisors associated with real eigenvalues of a matrix A and the signature of a Hermitian matrix H when AH is also Hermitian.We then obtain sets of equivalent conditions for a matrix to be similar to a real matrix; for a matrix to be symmetrizable; and for a matrix to be similar to a real diagonal matrix.As corollaries we obtain results on the eigenvalues and elementary divisors of products of two Hermitian matrices.Some of the results are not new; these are included to give a more complete survey of what is known in these particular areas, Recently a theorem on the stability of complex matrices, due to Lyapunov, has been generalized by Taussky [15,16], and independently, by Ostrowski and Schneider [12], Their result may be stated as follows: Given a complex matrix A 9 there exists a Hermitian H for which AH + HA* > 0 (positive definite) iί and only if A has no imaginary eigenvalues.Further, if AH + HA* > 0, the numbers of eigenvalues of A with positive and negative real parts equal respectively the numbers of eigenvalues of H which are positive and negative.Further generalizations of these results have been obtained by Schneider and this author [4, 6], under the condition that AH + HA* ^ 0 (positive semi-definite).This paper will use these results and methods to prove the theorems mentioned in the synopsis above.I wish to acknowledge with thanks the contribution of Professor Emilie Haynsworth, who pointed out to me the connection between [7] and my results, and thus sparked this investigation.I also wish to thank the referee for many helpful comments, and for references to several related papers, especially [13], [14], and [17], with which I had not been familiar.2* Definitions* We define the inertia of a complex matrix A to be In A = (π(A), v(A), δ(A)), where π(A), v(A), 3 (A) are respectively the number of eigenvalues of A with positive, negative, and zero real parts.We shall always let G, H and K represent Hermitian matrices; we denote the signature of H by σ(H) = π(H) -v(H).We shall define, as in [12], R(AH) = i(AH + HA*).