Explanation Given Information: A collection of n × n matrices is said to be a convex set, if whenever A and B are in the set and t is a number between 0 to 1, then t A + ( 1 − t ) B is also in the set. Formula used: For a square matrix, if the sum of the entries in each column is one and also the sum of the entries in each row is one, then the matrix is said to be doubly stochastic matrix. Proof: Assume A and B are n × n doubly stochastic matrices, A = [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋮ ⋮ a n 1 a n 2 ⋯ a n n ] n × n And, B = [ b 11 b 12 ⋯ b 1 n b 21 b 22 ⋯ b 2 n ⋮ ⋮ ⋮ ⋮ b n 1 b n 2 ⋯ b n n ] n × n Therefore, ∑ a i j = 1 and ∑ b i j = 1 . Now, consider a matrix C = t A + ( 1 − t ) B . Where, 0 < t < 1 . Therefore, C = t A + ( 1 − t ) B = t [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋮ ⋮ a n 1 a n 2 ⋯ a n n ] + ( 1 − t ) [ b 11 b 12 ⋯ b 1 n b 21 b 22 ⋯ b 2 n ⋮ ⋮ ⋮ ⋮ b n 1 b n 2 ⋯ b n n ] = [ t a 11 t a 12 ⋯ t a 1 n t a 21 t a 22 ⋯ t a 2 n ⋮ ⋮ ⋮ ⋮ t a n 1 t a n 2 ⋯ t a n n ] + [ ( 1 − t ) b 11 ( 1 − t ) b 12 ⋯ ( 1 − t ) b 1 n ( 1 <

Finite Mathematics & Its Applications (12th Edition)

12th Edition

ISBN: 9780134437767

Author: Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair

Publisher: PEARSON

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Chapter 8, Problem 7P

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To prove: The set of all n×n doubly stochastic matrices is a convex set.

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Chapter 8 Solutions

Finite Mathematics & Its Applications (12th Edition)

Ch. 8.1 - 1. Is a stochastic matrix? Ch. 8.1 - 2. Learning Process An elementary learning process...Ch. 8.1 - In Exercises 1-6, determine whether or not the...Ch. 8.1 - In Exercises 1-6, determine whether or not the...Ch. 8.1 - In Exercises 1-6, determine whether or not the...Ch. 8.1 - Prob. 4E Ch. 8.1 - In Exercises 1-6, determine whether or not the...Ch. 8.1 - Prob. 6E Ch. 8.1 - In Exercises 7–12, write a stochastic matrix...Ch. 8.1 - Prob. 8E

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To prove: The set of all n × n doubly stochastic matrices is a convex set.

Solution Summary: The author proves that the set of all ntimes n doubly stochastic matrices is a convex set.

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To prove: The set of all n×n doubly stochastic matrices is a convex set.

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Chapter 8 Solutions