W&M Optimal Gersgorin-Style Estimation Of The Largest Singular Value, Ii

. In estimating the largest singular value in the class of matrices equiradial with a given n -by- n complex matrix A , it was proved that it is attained at one of n ( n − 1) sparse nonnegative matrices (see C.R. Johnson, J.M. Pe˜na and T. Szulc, Optimal Gersgorin-style estimation of the largest singular value; Electronic Journal of Linear Algebra Algebra Appl. , 25:48–59, 2011). Next, some circumstances were identiﬁed under which the set of possible optimizers of the largest singular value can be further narrowed (see C.R. Johnson, T. Szulc and D. Wojtera-Tyrakowska, Optimal Gersgorin-style estimation of the largest singular value, Electronic Journal of Linear Algebra Algebra Appl. , 25:48–59, 2011). Here the cardinality of the mentioned set for n -by- n matrices is further reduced. It is shown that the largest singular value, in the class of matrices equiradial with a given n -by- n complex matrix, is attained at one of n ( n − 1) / 2 sparse nonnegative matrices. Finally, an inequality between the spectral radius of a 3-by-3 nonnegative matrix X and the spectral radius of a modiﬁcation of X is also proposed.

1. Introduction. Let M n (C) be the set of all n-by-n complex matrices. For a given matrix A = (a ij ) ∈ M n (C), we set P k (A) = j =k | a k,j |, k = 1, . . . , n, and define the class Λ(A) of matrices equiradial with A by where, for an X = (x i,j ) ∈ M n (C), D(X) = diag(x 1,1 , . . . , x n,n ). In particular, we will focus on the finite subset of Λ(A) that consists of n(n − 1) nonnegative matrices So, in particular, for a given 3-by-3 complex matrix A = (a ij ), the mentioned subset consists of the following 6 matrices: Remark 1. For our purposes, throughout this paper, it will be assumed that A has at least one nonzero off-diagonal entry in each row and that all its diagonal entries are nonzero. Then, it is easy to observe that any matrix A (s,k) (s, k ∈ {1, . . . , n} and s = k) has exactly 2n nonzero entries, i.e. all diagonal entries, all off-diagonal entries of the sth column and the (k, s)th entry.
We now describe how the matrices A (s,k) play an important role in estimation of the largest singular value among matrices equiradial with A (see [3]). Upper bounds for the largest singular value σ 1 (A) of a square matrix A in terms of possible simple functions of the entries of a matrix have many potential theoretical and practical applications. By simple functions we mean those which use "Gersgorin-type" data related to a matrix, i.e., diagonal entries and sums of the moduli of off-diagonal entries. Observe that for a given A ∈ M n (C) all matrices from Λ(A) share this type of information and therefore, from this point of view, they can be equal. So, the above comment motivates us to state the following question, which was studied in [3] and we refer to as the "motivating question": Given a matrix A, what is the maximum singular value among the matrices that are equivalent with A? I. e., where σ 1 (X) is the largest singular value of an n-by-n complex matrix X (equiradial with A). It was proved in [3] (Theorem 3) that one of the n(n − 1) matrices A (s,k) attains this maximum. Here, we show that the number n(n − 1) of candidates for a maximum can be reduced to n(n − 1)/2.

Results.
We start with the main result of the paper.
Theorem 2. Consider n × n matrices A (k,l) and A (l,k) and suppose that Then Proof. Without loss of generality, we set (k, l) = (1, 2). Applying the Perron-Frobenius theorem to the nonnegative matrix (A (2,1) ) T A (2,1) , we can deduce that there exists a nonnegative unit vector Let us consider two cases. First, we assume that x 2 > x 1 . Then where α is a positive number. So, by (3), (2) becomes which can be written as where P is the permutation matrix P = (e 2 , e 1 , e 3 , . . . , e n ) T with e i the i-th row of the identity n × n matrix. So, from (4), we obtain and, as P is a unitary matrix, we finally get (5) σ 2 1 (A (2,1) ) < σ 2 1 (A (1,2) ).
To prove the remaining case, assume that x 1 ≥ x 2 . Then we have (A (2,1) ).

Example 3. [3]. Let
be real polynomials such that their maximum modulus rootsx 1 andx 1 , respectively, are real and positive and let Then we havex 1 >x 1 .
Proof. It is easy to see that both f 1 and f 2 are increasing functions either for any , otherwise. Then, by (7) and the forms of f 1 and f 2 , we get f 2 (x 1 ) = a 3 − b 3 > 0 which, by the monotonicity of f 2 , completes the proof.
Theorem 5. Let A = (a i,j ) be a 3-by-3 nonnegative matrix and letÃ = (ã i,j ) be obtained from A by replacing an off-diagonal entry a i,j by a j,i and vice versa.
where ρ(X) denotes the spectral radius of a square matrix X.
Proof. We first observe that characteristic polynomials of A andÃ differ at most in the constant term. So, using the Perron-Frobenius theory [1], the assertion follows by applying Lemma 4. Then, since A is nonnegative with row sums 8, ρ(A) = 8 (see also [4]). In this case we haveÃ