152 Matrices and VectorsThis Section is a very concise introduction to the algebra of matrices and vectors.2.1 MatricesA (real) matrix of size m × n is an array of mn real numbers arranged in m rows and n columns:A =a11··· a1n......am1··· amn.The n × m matrix ATobtained by exchanging rows and columns of A is called the transpose ofA. A matrix A is said to be symmetric if A = AT.The sum of two matrices of equal size is the matrix of the entry-by-entry sums, and the scalarproduct of a real number a and an m ×n matrix A is the m ×n matrix of all the entries of A, eachmultiplied by a. The difference of two matrices of equal size A and B isA − B = A + (−1)B .The product of an m ×p matrix A and a p × n matrix B is an m × n matrix C with entriescij=pXk=1aikbkj.ExamplesLetA =3 0 −21 −1 0and B =2 −11 3.Then,AT=3 10 −1−2 0, BT=2 1−1 3andC = BA =2 · 3 + (−1) · 1 2 · 0 + (−1) · (−1) 2 · (−2) + (−1) · 01 · 3 + 3 · 1 1 · 0 + 3 · (−1) 1 · (−2) + 3 · 0=5 1 −46 −3 −2.The product AB is not defined, because A and B have incompatible sizes. Furthermore,3A =3 · 3 3 ·0 3 ·(−2)3 · 1 3 · (−1) 3 ·0=9 0 −63 −3 0,16 2 MATRICES AND VECTORSA+C =3 0 −21 −1 0+5 1 −46 −3 −2=3 + 5 0 + 1 −2 + (−4)1 + 6 −1 + (−3) 0 + (−2)=8 1 −67 −4 −2andA−C =3 0 −21 −1 0−5 1 −46 −3 −2=3 − 5 0 − 1 −2 − (−4)1 − 6 −1 − (−3) 0 − (−2)=−2 −1 2−5 2 2.2.2 VectorsA (real) n-dimensional vector is an n-tuple of real numbersv = (v1, . . . , vn) .There is a natural, one-to-one correspondence between n-dimensional vectors and n × 1 ma-trices:(v1, . . . , vn) ↔v1...vn.The matrix on the right is called the column vector corresponding to the vector on the left.There is also a natural, one-to-one correspondence between n-dimensional vectors and 1 × nmatrices:(v1, . . . , vn) ↔v1··· vn .The matrix on the right is called the row vector corresponding to the vector on the left.If a is a vector, then the symbol a also denotes the corresponding column vector, so that thecorresponding row vector is aT.All algebraic operations on vectors are inherited from the corresponding matrix operations,when defined. In addition, the inner product of two n-dimensional vectorsa = (a1, . . . , an) and b = (b1, . . . , bn)is the real number equal to the matrix product aTb. It is easy to verify that this is also equal tobTa. Two vectors that have a zero inner product are said to be orthogonal.The norm of a vector a iskak =√aTa ,obviously a nonnegative number. A unit vector is a vector with norm one.ExamplesThe vector a = (2, −1, 0) corresponds to row vector aT= [2, −1, 0] and to column vectora =2−10.17The inner product of a and b = (1, 0, −1) isaTb = [2, −1, 0]10−1= 2 · 1 + (−1) · 0 + 0 · (−1) = 2 ,and the norm of a iskak =√aTa =p2 · 2 + (−1) · (−1) + 0 · 0 =√5 ≈ 2.2361 .The vectorc =1√5a =2√5−1√50√5≈0.8944−0.44720has unit norm:kck =√cTc =s2√52+−1√52+0√52=r22+ (−1)2+ 025=r55=√1 = 1 .3 Linear, Deterministic, Stationary, Discrete Dynamic SystemsThe book shows examples of scalar, first order, dynamic systems of the forma(n + 1) = ca(n) , possibly with a constant input, a(n + 1) = ca(n) + uas well as systems of dynamic systems of the forma(n + 1) = Aa(n) + Bb(n)b(n + 1) = Ca(n) + Db(n) .In recitation, you saw scalar, second order, dynamic systems of the forma(n + 1) = 2ba(n) − ca(n − 1)and saw plots of their evolution.These three types of systems provide a rather rich repertoire of responses, and are all instantia-tions of deterministic, linear, stationary, discrete dynamic systems (without “scalar”). To keep themathematics simple, we limit ourselves to 2 ×2 systems, in which only at most two variables a(n)and b(n) appear.As we saw in the introduction, a dynamic system is a system with inputs and outputs that varyover time. A model of the system is a mathematical description of how the quantities of interestchange with time. The following are simple examples of a dynamic system:• An electric circuit, whose input is the current in a given branch and whose output is a voltageacross a pair of nodes.18 3 LINEAR, DETERMINISTIC, STATIONARY, DISCRETE DYNAMIC SYSTEMS• A chemical reactor, whose inputs are the external temperature, the temperature of the gasbeing supplied, and the supply rate of the gas. The output can be the temperature of thereaction product.• A mass suspended from a spring. The input is the force applied to the mass and the outputis the position of the mass.In all these examples, what is input and what is output is a choice that depends on the application.Also, all the quantities in the examples vary continuously with time. In other cases, as for instancefor switching networks and computers, it is more natural to consider time as a discrete variable. Iftime varies continuously, the system is said to be continuous; if time varies discretely, the system issaid to be discrete. Even if the system per se is continuous, it is possible to restrict its description toa discrete set of points in time, so a discrete model can describe a continuous system. This Sectionconsiders only discrete models of dynamic systems.Given a dynamic system, continuous or discrete, the modeling problem is to somehow correlateinputs (causes) with outputs (effects). The examples above suggest that the output at time t cannotbe determined in general by the value assumed by the input quantity at the same point in time.Rather, the output is the result of the entire history of the system. An effort of abstraction istherefore required, which leads to postulating a new quantity, called the state, which summarizesinformation about the past and the present of the system. Specifically, the value x(t) taken by thestate at time t must be sufficient to determine the output at the same point in time. Also, knowledgeof both x(t1) and u[t1,t2), that is, of the state at time t1and the input over the interval t1≤ t < t2,must allow computing the state (and hence the output) at time t2. For the mass attached to aspring, for instance, the state could be the position and velocity of the mass. In fact, the laws ofclassical mechanics allow computing the new position and velocity of