2 A (left) stochastic matrix is one which has only non-negative (> 0) entries, and such that the entries in each column sum to 1. Let A be any (that is, a general) 2×2 stochastic matrix. (a) Show that one of the eigenvalues of A is 1. (b) Show that a general nxn stochastic matrix also has one eigenvalue equal to 1. (c) What happens if the column sums are 1, as above, but the entries may be any (possibly negative) real numbers. Hint Consider adding all the rows together.
2 A (left) stochastic matrix is one which has only non-negative (> 0) entries, and such that the entries in each column sum to 1. Let A be any (that is, a general) 2×2 stochastic matrix. (a) Show that one of the eigenvalues of A is 1. (b) Show that a general nxn stochastic matrix also has one eigenvalue equal to 1. (c) What happens if the column sums are 1, as above, but the entries may be any (possibly negative) real numbers. Hint Consider adding all the rows together.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:2 A (left) stochastic matrix is one which has only non-negative (0) entries, and such that the entries in
each column sum to 1. Let A be any (that is, a general) 2×2 stochastic matrix.
(a) Show that one of the eigenvalues of A is 1.
(b) Show that a general nxn stochastic matrix also has one eigenvalue equal to 1.
(c) What happens if the column sums are 1, as above, but the entries may be any (possibly negative) real
numbers.
Hint Consider adding all the rows together.
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