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Definition B : A square nonnegative matrix, T, is primitive if there exists a positive inte-
ger k such that ?1 > 0.
Theorem B 1: If the n x n nonnegative matrix T is irreducible (Definition A7) and aper-
iodic (Definition A9), then T is primitive and conversely.
Theorem B2: Let T be an n x n nonnegative primitive matrix. Then there exists an eigen-
value r of T such that
(a) ris real, r > 0
(b) r has corresponding left and right eigenvectors with strictly positive
components
(c) r > I XI for any eigenvalue X # r
(d) the eigenvectors associated with r are unique to constant multiples
(e) If 0 5 B T and p is an eigenvalue of B, then | 11 < r. Moreover, I| p = r
implies B = T.
(f) r is a simple root of the characteristic polynomial of T.
Definition B2: The eigenvalue r asserted in Theorem B2 is called the Perron-Frobenius
eigenvalue of the nonnegative primitive matrix T.
Corollary B l: Let TW1 be the components of a nonnegative primitive matrix T having
Perron-Frobenius eigenvalue r. Then
min n Tj < r 5 max E T,
i j i J
with equality on either side implying equality throughout (i.e. r can only be equal to the
maximal or minimal row sum if all row sums are equal). A similar proposition holds for
column sums.