Bounds for the spectral radius of a matrix

Let A = an be an n X n matrix with complex entries. We define p (A) to be the spectral radius of A and | A | to be the matrix | a, y |. A. Brauer 1, W. Ledermann 2 and A. Ostrowski 4 have developed bounds for p (\ A |). Their results, coupled with the result of Perron and Frobenius 6 that p (A) ^ p (\ A |) give upper bounds for p (A) which are not less than p (\ A |). These bounds are restricted to matrices with nonzero entries and do not take into account the effect of the phases of the entries of A on p (A). In Section I of this paper we obtain a sequence of bounds for p (A) in terms of p (| Ar |) (r = 1, 2, ) which are less than or equal to p (| A |) and converge to p (A). In this manner we are partially accounting for the effect on p (A) of the phases of the a, y. In Section II we derive bounds for p (A) in terms of the Frobenius norm of A. These bounds always lie in the field of values of A, are computationally well suited to complex matrices and 'can be used in conjunction with the techniques of Section I. The authors are indebted to Olga Taussky and Alston Householder for suggestions. Lemma 2. The w* (fc = 1, 2, ) form a sequence of upper bounds for p (A) which converges to p (A). Proof. Since P (Ak) p (\), it follows that p{A) p{\ Ak \) m = *, which proves our first assertion. To prove convergence of the a>& we define the multiplicative matrix norm N{A) = max (23 I and use the general results 3 that limiV (A*) 1M= P{A) k-*oo and p{A) * g uk N{Ak). Taking fcth roots we conclude lim uk = p{A).