Abstract
Motivated by a work of Boros, Brualdi, Crama and Hoffman, we consider the sets of (i) possible Perron roots of nonnegative matrices with prescribed row sums and associated graph, and (ii) possible eigenvalues of complex matrices with prescribed associated graph and row sums of the moduli of their entries. To characterize the set of Perron roots or possible eigenvalues of matrices in these classes we introduce, following an idea of Al'pin, Elsner and van den Driessche, the concept of row uniform matrix, which is a nonnegative matrix where all nonzero entries in every row are equal. Furthermore, we completely characterize the sets of possible Perron roots of the class of nonnegative matrices and the set of possible eigenvalues of the class of complex matrices under study. Extending known results to the reducible case, we derive new sharp bounds on the set of eigenvalues or Perron roots of matrices when the only information available is the graph of the matrix and the row sums of the moduli of its entries. In the last section of the paper a new constructive proof of the Camion–Hoffman theorem is given.
Highlights
To characterize the set of Perron roots or possible eigenvalues of matrices in these classes we introduce, following an idea of Al’pin, Elsner and van den Driessche, the concept of row uniform matrix, which is a nonnegative matrix where all nonzero entries in every row are equal
The use of the row sums of a matrix to determine nonsingularity or to bound its spectrum has its origins in the 19th century [19, Section 2] and has led to a vast literature associated with the name of Geršgorin and his circles [22]
One of the first observations, due to Frobenius, was that the Perron root ρ(A) of a nonnegative matrix A ∈ Rn+×n is bounded by n min ri(A)
Summary
The use of the row sums of a matrix to determine nonsingularity or to bound its spectrum has its origins in the 19th century [19, Section 2] and has led to a vast literature associated with the name of Geršgorin and his circles [22]. Al’pin [2], Elsner and van den Driessche [12] sharpened the classical bounds of Frobenius by considering a matrix B which has the same zero-nonzero pattern as A, and whose entries are equal to the row sums of A in the corresponding rows We formalize this idea in the following definition. We do not prescribe the moduli of diagonal entries, and include these moduli in the row sums instead This allows us, in particular, to combine the problem statement of Boros, Brualdi, Crama and Hoffman [4] with that of Al’pin [2], Elsner and van den Driessche [12] and to generalize all above mentioned results removing the restriction that B is irreducible. Other proofs of the Camion–Hoffman theorem have been given by Levinger and Varga [17], and Engel [13]
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