Some spectral properties of an operator associated with a pair of nonnegative matrices

Author:
M. V. Menon

Journal:
Trans. Amer. Math. Soc. **132** (1968), 369-375

MSC:
Primary 15.60

DOI:
https://doi.org/10.1090/S0002-9947-1968-0225802-2

MathSciNet review:
0225802

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Abstract: An operator--in general nonlinear--associated with a pair of non-negative matrices, is defined and some of its spectral properties studied. If the pair of matrices are a square matrix *A* and the identity matrix of the same order, the operator reduces to the linear operator *A*. The results obtained include generalizations of one of the principal conclusions of the theorem of Perron-Frobenius.

**[1]**R. A. Brualdi, S. V. Parter and H. Schneider,*The diagonal equivalence of a non-negative matrix to a stochastic matrix*, J. Math. Anal. Appl.**16**(1966), 31-50. MR**0206019 (34:5844)****[2]**F. R. Gantmacher,*The theory of matrices*. II, Chelsea, New York, 1959.**[3]**M. V. Menon,*Reduction of a matrix with positive elements to a doubly stochastic matrix*, Proc. Amer. Math. Soc.**18**(1967), 244-247. MR**0215873 (35:6708)****[4]**-,*Matrix links, an extremisation problem and the reduction of a non-negative matrix to one with prescribed row and column sums*, Canad. J. Math.**20**(1968), 225-232. MR**0220752 (36:3804)****[5]**M. Morishima,*Generalizations of the Frobenius-Wielandt theorems for non-negative square matrices*, J. London Math. Soc.**36**(1961), 211-220. MR**0124347 (23:A1661)****[6]**R. Sinkhorn and P. Knopp,*Concerning non-negative matrices and doubly stochastic matrices*, Pacific J. Math.**21**(1967), 343-348. MR**0210731 (35:1617)**

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DOI:
https://doi.org/10.1090/S0002-9947-1968-0225802-2

Article copyright:
© Copyright 1968
American Mathematical Society