Question

Question 4 [35 marks in totalj An n x n matrix A is called a stochastic matrix if it! satisfies two conditions: (i) all entries of A are non-negative; and (ii) the sum of entries in each column is one. If the (,) entry of A is denoted by any for ij € {1, 2,...,n}, then A is a stochastic matrix when alij 20 for all i and j and I j = 1 for all j. These matrices are useful because each n x n stochastic matrix describes a Markov chain on the state space {1,2,...,n}. Even though all entries of a stochastic matrix are real, for studying its powers, we are also interested in its complex eigenvalues. Throughout this question, you are not allowed to use Brouwer's fixed point theorem even if you happen to know it. 1. (5 marks] (Straight forward) We first prove a result on complex number algebra which may be useful in later parts: for any positive integer n and complex numbers 21, ..., n 21 +...+2n = 1 + ... + 2n. (Hint: we know that this inequality is true when n = 2 and you can use this fact without proof.) 2. (5 marks) (Straight forward) Let An-1 = {(21,22,...,n) ER":1;> 0 for all j and wi+ 22 + ... + In = 1}. In words, An-1 consists of n-dimensional vectors whose compo- nents are non-negative and add up to one. Each member of An-1 can be interpreted as a probability mass function on the state space {1,2,...,n}. Show that if A is a stochastic matrix, then Av E An-1 for every ve An-1. 3. [5 marks] (Straight forward) Using the fact that a square matrix and its transpose have the same determinant, show that A and AT have the same set of complex) eigenvalues.

Answer 1

ANSWER:- We know that inequality is toue cohen 0-2 and 121+22+.. +2n) < 1211+1221 +...+|2n|| Now, Nothing there to prove for na,12,1812,1 For n=8, [ 2=2+iy 2= {(2+2) 2 Rect) = zt 17,+2012 = (21+29) •12,+zo) - (Z1+2g) (E1+2g) = 213 +Zn +21 79 +297 • 13,+17212+2 RE (2122) $ 12,12+13,12+2 17,1781 < 121272/211 -17al +12014 - (12/+1281)2 = 1217 2218 < 1211+12a) fod (nt) complex numbers 21179 ...2n4 That is, 1244 --2n+ 1 s 1217-- Iznel then lzitzg ---+201 = |21+92 +-- Inu)+211 < 121tz&+ -- Zn-iltlen! €1211+122)+. - 13n+1+1zn1

That is, 12+..+20 |<lzitt.. + lenl for any complex numbers @j,79..Zn positive intergen n ond let An-= $190,22 -- XN) ER":%;* j and X1+22+..xn=1} Here Any: n-dimensional vectors cohose components are non- negative let A-lagpxn is stochastic matox then Q9f 70-* igje $1,8.-OS Qiz 41 * 3+12. let v: (1988 - • Xn De ANI There fore , 6770 for j=1,2,...n and 21+342 --- +2n=1 If Av = (You YQ - YO) then You Že xj for i1,...n Here aj 20 and 2j70 V inje svQ...)3 So, y;7,0 for i=1,2,..n 942.. 49 070979., st. tapos cho WI Wisa (303)2

so, Anny : 14,, 40..Yoe Aney ise AV € Ant for every WE ANI 6 let X is a ligen value Of A det (A-AIN) = 0 det (AT-I In) det (A-19027 det (A - AIN) - AT then ^ is an eigen value of is eigen value of AT 110 s is an eigen value of APPA A and AT is both have some set of eigen values *** - - Thanking you please give like it helps to motivate me to do more work friend