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The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: a survey. (English) Zbl 0613.15017
The author gives a survey, beginning with Frobenius, of certain properties of a nonnegative matrix which are related to, or can be expressed by, properties of the marked reduced graph of the matrix. For a nonnegative square matrix P with Perron-Frobenius root \(\rho\) the areas surveyed are: (1) positivity properties of the eigenvectors and generalized eigenvectors of P; (2) the nonnegative solutions x of \((\lambda I-P)x=b\), for a given nonnegative vector b and given real number \(\lambda\) ; (3) the Jordan blocks associated with \(\rho\) in the Jordan canonical form of P; (4) the growth of the elements of \(P^ m\) as m grows.
Reviewer: F.J.Gaines

15B48 Positive matrices and their generalizations; cones of matrices
15A21 Canonical forms, reductions, classification
15A18 Eigenvalues, singular values, and eigenvectors
Full Text: DOI
[1] Frobenius, G.F.; Frobenius, G.F., Über matrizen aus nict negativen elementen, (), 546-567
[2] Schneider, H., Matrices with non-negative elements, ()
[3] Schneider, H., An inequality for latent roots applied to determinants with dominant principal diagonal, J. London. math. soc., 28, 8-20, (1953) · Zbl 0050.01103
[4] Schneider, H., The elementary divisors associated with 0 of a singular M-matrix, Proc. Edinburgh math. soc., 10, 2, 108-122, (1956) · Zbl 0074.25401
[5] Gantmacher, F.R., The theory of matrices, (1959), Moscow, 1953 · Zbl 0085.01001
[6] Carlson, D.H., A note on M-matrix equations, SIAM j., 11, 213-217, (1963)
[7] Cooper, C.D.H., On the maximum eigenvalue of a reducible non-negative real matrix, Math. Z., 13, 213-217, (1973) · Zbl 0261.15006
[8] Rothblum, U.G., Algebraic eigenspaces of non-negative matrices, Linear algebra appl., 12, 281-292, (1975) · Zbl 0321.15010
[9] Richman, D.; Schneider, H., On the singular graph and the Weyr characteristic of an M-matrix, Aequationes math., 17, 208-234, (1978) · Zbl 0379.15009
[10] Rothblum, R.G., A rank characterization of the number of final classes of a nonnegative matrix, Linear algebra appl., 23, 65-68, (1979) · Zbl 0398.15015
[11] Friedland, S.; Schneider, H., The growth of powers of a non-negative matrix, SIAM J. algebraic discrete methods, 1, 185-200, (1980) · Zbl 0498.65018
[12] Rothblum, U.G., Sensitive growth analysis and multiplicative systems, I the dynamic approach, SIAM J. algebraic discrete methods, Yale univ. report, 2, 25-34, (1977) · Zbl 0498.60047
[13] Victory, H.D., On nonnegative solutions to matrix equations, SIAM J. algebraic discrete methods, 6, 406-412, (1985) · Zbl 0586.15003
[14] Artzroumi, M., On the asymptotic behavior of powers of nonnegative matrices, presented at annual meeting, Population assoc. amer., (1986)
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