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Understanding Matrices, Digraphs, Markov Chains & Their Application

Explore the foundational concepts of matrices, digraphs, and Markov chains. Understand how matrices are used to solve equations and how Markov chains model transitions. Learn about transition probabilities and setting up matrices for real-world problems.

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Understanding Matrices, Digraphs, Markov Chains & Their Application

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  1. Matrices, Digraphs, Markov Chains & Their Use

  2. Introduction to Matrices • A matrix is a rectangular array of numbers • Matrices are used to solve systems of equations • Matrices are easy for computers to work with

  3. Matrix arithmetic • Matrix Addition • Matrix Multiplication

  4. Introduction to Markov Chains • At each time period, every object in the system is in exactly one state, one of 1,…,n. • Objects move according to the transition probabilities: the probability of going from state j to state i is tij • Transition probabilities do not change over time.

  5. The transition matrix of a Markov chain • T = [tij] is an nn matrix. • Each entry tij is the probability of moving from state j to state i. • 0 tij  1 • Sum of entries in a column must be equal to 1 (stochastic).

  6. Example:Customers can choose from a major Long Distance carrier (SBC) or others ores: • Each year 30% of SBC customers switch to other carrier, while 40% of other carrier switch to SBC. • Set Up the matrix for this Problem

  7. Example:The transition matrix in 2ndand 3rd year..

  8. How many SBC customers will be there 2 years from now? How many SBC customers will be there 3 years from now?

  9. How many non-SBC customers will be there 2 years from now? • How many non SBC customers will be there 3 years from now?

  10. Thank you!

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