Nonnegative Matrices and Applicable Topics in Linear Algebra by Alexander Graham

Author:Alexander Graham
Language: eng
Format: epub
Publisher: Courier Publishing
Published: 2019-03-09T16:00:00+00:00

then

so that x is an eigenvector corresponding to the eigenvalue λ of A.

(4)Next we must reconsider whether a maximum value of defined in (4.1) exists for irreducible matrices. We proceed as in Section 4.2, and consider

and

Whereas for a positive matrix A, for an irreducible matrix it is possible that

So is not necessarily continuous on S.

We can overcome this difficulty by defining the set

So that

Where (for all i), so that is continuous on the closed and bounded set Q.

It follows that attains a maximum value for some (say).

For each y ∊ S,

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