Let A be an invertible stochastic matrix. (a) Show that e¹A−¹ = eT. (This proves that the columns of A-¹ each sum to 1, and hence ||A-¹x||1 = 1 whenever ||x||1 = 1.) Hint: Start with e¹ A¯¹, and use what you know about A to substitute an expression for eT. (b) Nevertheless, it's not true in general that the inverse of an invertible stochastic matrix is stochastic. Give an example of an invertible 2×2 matrix A such that A-1 is not stochastic.
Let A be an invertible stochastic matrix. (a) Show that e¹A−¹ = eT. (This proves that the columns of A-¹ each sum to 1, and hence ||A-¹x||1 = 1 whenever ||x||1 = 1.) Hint: Start with e¹ A¯¹, and use what you know about A to substitute an expression for eT. (b) Nevertheless, it's not true in general that the inverse of an invertible stochastic matrix is stochastic. Give an example of an invertible 2×2 matrix A such that A-1 is not stochastic.
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:Let A be an invertible stochastic matrix.
(a) Show that e A−¹ = e. (This proves that the columns of A-1 each sum to 1, and hence
||A-¹x||1 = 1 whenever ||x||1 = 1.)
Hint: Start with e²A-¹, and use what you know about A to substitute an expression for
e™.
(b) Nevertheless, it's not true in general that the inverse of an invertible stochastic matrix is
stochastic. Give an example of an invertible 2×2 matrix A such that A-¹ is not stochastic.
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