×

A function about the largest eigenvalue of a nonnegative matrix. (Chinese. English summary) Zbl 0840.15007

For a nonnegative matrix \(A_{n \times n} = (a_{ij})\), let \(A (\beta)_{n \times n} = (a^\beta_{ij})\) be another nonnegative matrix for any \(\beta \geq 0\). The largest eigenvalue of \(A (\beta)\) as a real function of \(\beta\) is called the eigenfunction of \(A\) and denoted by \(\Phi_A (\beta)\). In this article, some elementary properties of \(\Phi_A\) are demonstrated through simple discussion, and hence lower and upper bounds of \(\alpha\) satisfying \(\Phi_A (\alpha) = 1\) are obtained. Particularly, the value of this \(\alpha\) is perfectly determined for a class of nonnegative matrices called equiratio contraction, and this result is convenient for the computation of the Hausdorff dimension of some fractals.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
28A80 Fractals
28A78 Hausdorff and packing measures
15B48 Positive matrices and their generalizations; cones of matrices
PDFBibTeX XMLCite