Page 102 - Textos de Matemática Vol. 39

92 CHARLES R. JOHNSON AND YULIN ZHANG
Theorem 1.11. Let A be an n-by-n nonsingular nonscalar matrix and suppose that β1, ..., βn, γ1, ...γn ∈ C are given complex numbers, repetitions allowed, so
that n
βiγi =detA.
i=1
Then, there exist nonderogatory matrices B with eigenvalues β1, ..., βn and C
with eigenvalues γ1, ...γn such that
A = BC.
Proof. In the event that neither the β’s nor the γ’s include repetitions, there is nothing to prove. If repetitions do occur and the β’s and γ’s may be ordered so that repeats among the β’s occur consecutively and among the γ’s occur consecutively, and A is not exceptional for β1γ1, ..., βnγn, then the factorization is guaranteed by Theorem 1.5. If A is exceptional for any pair of consecutive orderings, then the result follows from Lemma 1.10.
References
[1] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ.Press, 1991. [2] A. Sourour, A factorization theorem for matrices, Linear Multilinear 19 (1986), 141-147.
Dept. of Mathematics
College of William and Mary Williamsburg
VA 23187, USA
E-mail address: crjohnso@math.wm.edu
Centro de Matema´tica Universidade do Minho
4710 Braga, Portugal
E-mail address: zhang@math.uminho.pt