Understanding Mechanical Inverses and Transition Matrices in
School
University of California, Berkeley**We aren't endorsed by this school
Course
EECS 16A
Subject
Computer Science
Date
Dec 12, 2024
Pages
2
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Last Updated: 2023-09-15 16:061EECS 16ADesigning Information Devices and Systems IFall 2023Discussion 4A1. Mechanical InversesA square matrixAis said to be invertible if there exists a matrixBsuch thatAB=IandBA=I. The matrixBis called the inverse ofAand is denoted byA−1.Some properties of the inverse are:•(A−1)−1=A•(AT)−1= (A−1)TIn each part, determine whether the inverse ofAexists. If it exists, find it.(a)A=1009(b)A=abcd(c) (PRACTICE)A=5411(d)A=5515224114(e) (PRACTICE)A=5515224104UCB EECS 16A, Fall 2023, Discussion 4A, All Rights Reserved. This may not be publicly shared without explicit permission.1
Last Updated: 2023-09-15 16:0622. Transition MatrixSuppose we have a network of pumps as shown in the diagram below. Let us describe the state ofAandBusing a state vector⃗x[n] =xA[n]xB[n]wherexA[n]andxB[n]are the states at time-stepn.AB34121412(a) Find the state transition matrixS, such that⃗x[n+1] =S⃗x[n].Separately find the sum of the terms for each column vector inS. Do you notice any pattern?(b) Let us now find the matrixS−1such that we can recover the previous state⃗x[n−1]from⃗x[n].Specifically, solve forS−1such that⃗x[n−1] =S−1⃗x[n].(c) Now draw the state transition diagram that corresponds to theS−1that you just found.Also find the sum of the terms for each column vector inS−1. Do you notice any pattern?(d) Redraw the diagram from the first part of the problem, but now with the directions of the arrows re-versed. Let us call the state transmission matrix of this "reversed" state transition diagramT. DoesT=S−1?(e) Suppose we start in the state⃗x[1] =1212. Compute the state vector after 2 time-steps⃗x[3].(f)(Challenge practice problem)Given our starting state from the previous problem, what happens if welook at the state of the network after a lot of time steps? Specifically which state are we approaching,as defined below?⃗xfinal=limn→∞⃗x[n]Note that the final state needs to be what we call asteady state, meaningS⃗xfinal=⃗xfinal.Also what can you say aboutxA[n]+xB[n]?Use information from both of these properties to write out a new system of equations and solve for⃗xfinal.UCB EECS 16A, Fall 2023, Discussion 4A, All Rights Reserved. This may not be publicly shared without explicit permission.2