Remark equilibrium points often have interesting

Remark. Equilibrium points often have interesting interpretations. For example, if the
linear dynamical system describes the population dynamics of a country, with the vector c
denoting immigration (emigration when entries of c are negative), an equilibrium point is
a population distribution that does not change, year to year. In other words, immigration
exactly cancels the changes in population distribution caused by aging, births, and deaths.

Exercises 175
9.4 Reducing a Markov model to a linear dynamical system. Consider the 2-Markov model
x
t +1
= A
1
x
t
+ A
2
x
t - 1
, t = 2 , 3 , . . . ,
where x
t
is an n -vector. Define z
t
= ( x
t
, x
t - 1
). Show that z
t
satisfies the linear dynamical
system equation z
t +1
= Bz
t
, for t = 2 , 3 , . . . , where B is a (2 n ) ⇥ (2 n ) matrix. This idea
can be used to express any K -Markov model as a linear dynamical system, with state
( x
t
, . . . , x
t - K +1
).
9.5 Fibonacci sequence. The Fibonacci sequence y
0
, y
1
, y
2
, . . . starts with y
0
= 0, y
1
= 1, and
for t = 2 , 3 , . . . , y
t
is the sum of the previous two entries, i.e. , y
t - 1
+ y
t - 2
. (Fibonacci
is the name used by the 13th century mathematician Leonardo of Pisa.) Express this
as a time-invariant linear dynamical system with state x
t
= ( y
t
, y
t - 1
) and output y
t
,
for t = 1 , 2 , . . . . Use your linear dynamical system to simulate (compute) the Fibonacci
sequence up to t = 20. Also simulate a modified Fibonacci sequence z
0
, z
1
, z
2
, . . . , which
starts with the same values z
0
= 0 and z
1
= 1, but for t = 2 , 3 , . . . , z
t
is the di ↵ erence of
the two previous values, i.e. , z
t - 1
- z
t - 2
.
9.6 Recursive averaging. Suppose that u
1
, u
2
, . . . is a sequence of n -vectors. Let x
1
= 0, and
for t = 2 , 3 , . . . , let x
t
be the average of u
1
, . . . , u
t - 1
, i.e. , x
t
= ( u
1
+ · · · + u
t - 1
) / ( t - 1).
Express this as a linear dynamical system with input, i.e. , x
t +1
= A
t
x
t
+ B
t
u
t
, t = 1 , 2 , . . .
(with initial state x
1
= 0). Remark. This can be used to compute the average of an
extremely large collection of vectors, by accessing them one-by-one.
9.7 Complexity of linear dynamical system simulation. Consider the time-invariant linear
dynamical system with n -vector state x
t
and m -vector input u
t
, and dynamics x
t +1
=
Ax
t
+ Bu
t
, t = 1 , 2 , . . . . You are given the matrices A and B , the initial state x
1
, and the
inputs u
1
, . . . , u
T - 1
. What is the complexity of carrying out a simulation, i.e. , computing
x
2
, x
3
, . . . , x
T
? About how long would it take to carry out a simulation with n = 15,
m = 5, and T = 10 5 , using a 1 Gflop/s computer?

Chapter 10
Matrix multiplication
In this chapter we introduce matrix multiplication, a generalization of matrix-vector
multiplication, and describe several interpretations and applications.
10.1 Matrix-matrix multiplication
It is possible to multiply two matrices using matrix multiplication . You can multiply
two matrices A and B provided their dimensions are compatible , which means the
number of columns of A equals the number of rows of B . Suppose A and B are
compatible, e.g. , A has size m ⇥ p and B has size p ⇥ n . Then the product matrix
C = AB is the m ⇥ n matrix with elements
C
ij
=
p X
k =1
A
ik
B
kj
= A
i 1
B
1 j
+ · · · + A
ip
B
pj
, i = 1 , . . . , m, j = 1 , . . . , n. (10.1)
There are several ways to remember this rule. To find the i, j element of the
product C = AB , you need to know the i th row of A and the j th column of B .