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Stochastic Matrix Maths for Signals and Systems Linear Algebra in Engineering Lecture 12, Friday 6th November 2015 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON Stochastic matrices • Consider a matrix 퐴 with the following properties: All entries are positive and real. The elements of each column or each row or both each column and each row add up to 1. Based on the above a matrix that exhibits the above properties will have all entries ≤ 1. It is square. • This is called a stochastic matrix. • Stochastic matrices are also called Markov, probability, transition, or substitution matrices. • The entries of a stochastic matrix usually represent a probability. • Stochastic matrices are widely used in probability theory, statistics, mathematical finance and linear algebra, as well as computer science and population genetics. Stochastic matrices. Types. • There are several types of stochastic matrices: . A right stochastic matrix is a matrix of nonnegative real entries, with each row’s elements summing to 1. A left stochastic matrix is a matrix of nonnegative real entries, with each column’s elements summing to 1. A doubly stochastic matrix is a matrix of nonnegative real entries with each row’s and each column’s elements summing to 1. • A stochastic matrix often describes a so called Markov chain 푋푡 over a finite state space 푆. • Generally, an 푛 × 푛 stochastic matrix is related to 푛 “states”. • If the probability of moving from state 푖 to state 푗 in one time step is 푃푟 푗 푖 = [푝푖푗], th th the stochastic matrix 푃 is given by using 푝푖푗 as the 푖 row and 푗 column element: 푝11 푝12 … 푝1푛 푝21 푝22 … 푝2푛 푃 = ⋮ ⋮ ⋱ ⋮ 푝푛1 푝푛2 … 푝푛푛 • Depending on the particular problem the above matrix can be formulated in such a way so that is either right or left stochastic. Products of stochastic matrices. Stochastic vectors. • An example of a left stochastic matrix is the following: 0.1 0.01 0.3 퐴 = 0.2 0.99 0.3 0.7 0 04 • You can prove that if 퐴 and 퐵 are stochastic matrices of any type then 퐴퐵 is also a stochastic matrix of the same type. • Consider two left stochastic matrices 퐴 and 퐵 with elements 푎푖푗 and 푏푖푗 respectively, and 퐶 = 퐴퐵 with elements 푐푖푗. • Let us find the sum of the elements of the 푗th column of 퐶: 푛 푛 푛 푛 푛 푛 푛 푖=1 푐푖푗 = 푖=1 푘=1 푎푖푘푏푘푗 = 푘=1 푖=1 푎푖푘푏푘푗 = 푘=1 푏푘푗 푖=1 푎푖푘 = 1 ∙ 1 = 1 • Based on the above the power of a stochastic matrix is a stochastic matrix. • I am interested in the eigenvalues and eigenvectors of a stochastic matrix. • Let’s call a vector with real, nonnegative entries 푝푘, for which all the 푝푘 add up to 1, a stochastic vector. For a stochastic matrix, every column or row or both is a stochastic vector. Stochastic matrices and their eigenvalues • I would like to prove that 휆 = 1 is always an eigenvalue of a stochastic matrix. • Consider again a left stochastic matrix 퐴. • Since the elements of each column of 퐴 add up to 1, the elements of each row of 퐴푇 should add up to 1. Therefore: 1 1 1 퐴푇 1 = 1 = 1 ∙ 1 ⋮ ⋮ ⋮ 1 1 1 • Therefore, 1 is an eigenvalue of 퐴푇 . The eigenvalues of 퐴푇 are obtained through the equation det 퐴푇 − 휆퐼 = 0. But: det 퐴푇 − 휆퐼 = det 퐴푇 − 휆퐼푇 = det 퐴 − 휆퐼 푇 = det (퐴 − 휆퐼) • Hence, the eigenvalues of 퐴 and 퐴푇 are the same, which implies that 1 is also an eigenvalue of 퐴. • Since det 퐴 − 퐼 = 0, the matrix 퐴 − 퐼 is singular, which means that there is a vector 푥 for which 퐴 − 퐼 푥 = 0 ⇒ 퐴푥 = 푥 • A vector of the null space of 퐴 − 퐼 is the eigenvector of 퐴 that corresponds to eigenvalue 휆 = 1. Stochastic matrices and their eigenvalues cont. • I would like now to prove that the eigenvalues of a stochastic matrix have magnitude smaller or equal to 1. 푣1 푣2 • Assume that 푣 = ⋮ is an eigenvector of a stochastic matrix 퐴 with an 푣푛 푛 푛 푛 푛 푛 eigenvalue 휆 > 1. Then 퐴 푣 = 휆 푣 implies 푗=1[퐴 ]푖푗푣푗 = 휆 푣푖 • 휆푛 has exponentially growing length for 푛 ⇢ ∞. The maximum value that 푛 푛 푛 푛 푛 [퐴 ]푖푗푣푗 can take is 푣max ∙ max 퐴 푖푗 ≤ 푣max ∙ 푛 since the entries of 퐴 푗=1 푖 푗=1 are ≤ 1 . 푛 푛 푛 • The relationship 푗=1[퐴 ]푖푗푣푗 = 휆 푣푖 must be valid for any 푛. There is always a 푛 푛 푛 minimum 푛 for which 푗=1[퐴 ]푖푗푣푗 ≤ 푣max∙ 푛 < 휆 푣푖 if 휆 > 1. • Based on the above, the assumption of an eigenvalue being larger than 1 can not be valid. Application of stochastic matrices 푘 • Consider again the system described by an equation of the form 푢푘 = 퐴 푢0, where 퐴 is now a stochastic matrix. 푘 푘 푘 • Previously, we managed to write 푢푘 = 퐴 푢0 = 푐1휆1 푥1 + 푐2휆2 푥2 + ⋯ where 휆푖 and 푥푖 are the eigenvalues and eigenvectors of matrix 퐴, respectively. • Note that the above relationship requires a complete set of eigenvectors. • If 휆1 = 1 and 휆푖 < 1, 푖 > 1 then the steady state of the system is 푐1푥1 (which is part of the initial condition 푢0). • I will use an example where 퐴 is a 2 × 2 matrix. Generally, an 푛 × 푛 stochastic matrix is related to 푛 “states”. Assume that the 2 “states” are 2 UK cities. • I take London and Oxford. I am interested in the population of the two cities and how it evolves. • I assume that people who inhabit these two cities move between them only. 푢ox 0.9 0.2 푢ox = 푢lon 푡=푘+1 0.1 0.8 푢lon 푡=푘 • It is now obvious that the column elements are positive and also add up to 1 because they represent probabilities. Application of stochastic matrices (cont.) 푢ox 0 푢ox 0.9 0.2 0 200 • I assume that = . Then = = . 푢lon 푡=푘=0 1000 푢lon 푘=1 0.1 0.8 1000 800 Problem: What is the population of the two cities after a long time? Solution: 0.9 0.2 Consider the matrix . The eigenvalues are 휆 = 1 and 휆 = 0.7. 0.1 0.8 1 2 (Notice that the second eigenvalue is found by the trace of the matrix.) 2 −1 The eigenvectors of this matrix are 푥 = and 푥 = 1 1 2 1 푢ox 푘 2 푘 −1 2 푘 −1 = 푐1휆1 + 푐2휆2 = 푐1 + 푐20.7 푢lon 푘 1 1 1 1 푢ox 0 I find 푐1, 푐2 from the initial condition = 푢lon 푡=푘=0 1000 0 2 −1 1000 2000 =푐 + 푐 and therefore, 푐 = and 푐 = 1000 1 1 2 1 1 3 2 3 Application of stochastic matrices (cont.) • Stochastic models facilitate the modeling of various real life engineering applications. • An example is the modeling of the movement of people without gain or loss: total number of people is conserved. .
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