LINEAR ALGEBRA Lemma 2.8 Suppose A=A*>0 and B=B>0. Then A>B iff P(BA-)<1 Proof. Since A>0, we have A>B iff 0<I-A-1/2BA-1/2 i.e. iff p(A-1/2BA-1/2)< 1. However, A-1/BA-1/2 and BA-1 are similar. hence P(BA-1)=P(A-1/BA-1/2)and the claim follow 2.8 Notes and references A very extensive treatment of most topics in this chapter can be found in Brogan 1991] Horn and Johnson[(1990, 1991]and Lancaster and Tismenetsky [1985. Golub and Van Loan s book [1983 contains many numerical algorithms for solving most of the problems in this chapter 2.9 Problems Problem 2.1 Let 10 A=211 101 Determine the row and column rank of A and find bases for Im(A), Im(A*), and Ker(A) Problem 2.2 Let Do=2 5. Find a D such that D*D=I and ImD=ImDo Furthermore, find a DI such that[D Di is a unitary matrix. Problem 2.3 Let A be a nonsingular matrix and a, yE C". Show Azy*A and det a Problem 2.4 Let A and b be compatible matrices. Show B(I+AB)-1=(I+BA)-1B,(I+A)-1=I-A(1+A)-1
24 LINEAR ALGEBRA Lemma 2.8 Suppose A = A∗ > 0 and B = B∗ ≥ 0. Then A>B iff ρ(BA−1) < 1. Proof. Since A > 0, we have A>B iff 0 < I − A−1/2BA−1/2 i.e., iff ρ(A−1/2BA−1/2) < 1. However, A−1/2BA−1/2 and BA−1 are similar, hence ρ(BA−1) = ρ(A−1/2BA−1/2) and the claim follows. ✷ 2.8 Notes and References A very extensive treatment of most topics in this chapter can be found in Brogan [1991], Horn and Johnson [1990, 1991] and Lancaster and Tismenetsky [1985]. Golub and Van Loan’s book [1983] contains many numerical algorithms for solving most of the problems in this chapter. 2.9 Problems Problem 2.1 Let A = 110 101 211 101 202 . Determine the row and column rank of A and find bases for Im(A), Im(A∗), and Ker(A). Problem 2.2 Let D0 = 1 4 2 5 3 6 . Find a D such that D∗D = I and ImD = ImD0. Furthermore, find a D⊥ such that D D⊥ is a unitary matrix. Problem 2.3 Let A be a nonsingular matrix and x,y ∈ Cn. Show (A−1 + xy∗) −1 = A − Axy∗A 1 + y∗Ax and det(A−1 + xy∗) −1 = det A 1 + y∗Ax. Problem 2.4 Let A and B be compatible matrices. Show B(I + AB) −1 = (I + BA) −1B, (I + A) −1 = I − A(I + A) −1.
2. 9. Problems Problem 2.5 Find a basis for the maximum dimensional stable invariant subspace of with 1.A nd Q=0 d 3.A=0,R 12 and Q=12 Problem 2.6 Let A=[aii. Show that a(A): =maxi,] lain defines a matrix norm. Give examples so that a(A)<p(A) and a(AB)>a(A)a(B) Problem 2.7 Let A=(41-1 and B=(1).(a) Find all r such that Az=B (b) Find the minimal norm solution r: min z: AT=B Problem 2.8 Let A .2 -5 and B= 4. Find an r such that llAr-B is minimized Problem 2.9 Let A< 1. Show (I-4)-1=I+A+42+ 2.|(-4)-1≤1+|A|+|42+ 3.|(I-41≥xx Problem 2.10 Let A e cmxn. Show that A|2≤‖Ale≤√hn‖A|2 Vh llz s lls vmlllz A|≤‖41≤m‖Al‖l Problem211LetA=xy’andx,y∈C". Show that42=A‖l=l‖xl‖lyl Problem 2.12 Let a 11+3. Find A] and a BE C2 such that A=BB
2.9. Problems 25 Problem 2.5 Find a basis for the maximum dimensional stable invariant subspace of H = A R −Q −A∗ with 1. A = −1 2 3 0 , R = −1 −1 −1 −1 , and Q = 0 2. A = 0 1 0 2 , R = 0 0 0 −1 , and Q = 1 2 2 4 3. A = 0, R = 1 2 2 5 , and Q = I2. Problem 2.6 Let A = [aij ]. Show that α(A) := maxi,j |aij | defines a matrix norm. Give examples so that α(A) < ρ(A) and α(AB) > α(A)α(B). Problem 2.7 Let A = 12 3 4 1 −1 and B = 0 1 . (a) Find all x such that Ax = B. (b) Find the minimal norm solution x: min {kxk : Ax = B}. Problem 2.8 Let A = 1 2 −2 −5 0 1 and B = 3 4 5 . Find an x such that kAx − Bk is minimized. Problem 2.9 Let kAk < 1. Show 1. (I − A)−1 = I + A + A2 + ··· . 2. (I − A)−1 ≤ 1 + kAk + kAk2 + ··· = 1 1−kAk . 3. (I − A)−1 ≥ 1 1+kAk . Problem 2.10 Let A ∈ Cm×n. Show that 1 √m kAk2 ≤ kAk∞ ≤ √n kAk2 ; 1 √n kAk2 ≤ kAk1 ≤ √m kAk2 ; 1 n kAk∞ ≤ kAk1 ≤ m kAk∞ . Problem 2.11 Let A = xy∗ and x,y ∈ Cn. Show that kAk2 = kAkF = kxk kyk. Problem 2.12 Let A = 1 1+ j 1 − j 2 . Find A1 2 and a B ∈ C2 such that A = BB∗
LINEAR ALGEBRA Problem 2.13 Let p P1P12 =P2P2|≥0 with PI∈cx.Show(P)≥ (B1),1≤i≤k Problem214Letx=X≥0 be partitioned as X=「xnX12 Ker X22 C Ker X12;(b)let X22= U2diag(A1, 0)U> be such that A1 is nonsingular and define X22: U2diag(Al, O)U(the pseudoinverse of X22); then show that Y= X12 X22 solves Y X22= X12; and(c)show that X11X1 T X12X X11-X12XX120 X2X12
26 LINEAR ALGEBRA Problem 2.13 Let P = P∗ = P11 P12 P∗ 12 P22 ≥ 0 with P11 ∈ Ck×k. Show λi(P) ≥ λi(P11), ∀ 1 ≤ i ≤ k. Problem 2.14 Let X = X∗ ≥ 0 be partitioned as X = X11 X12 X∗ 12 X22 . (a) Show KerX22 ⊂ KerX12; (b) let X22 = U2diag(Λ1, 0)U∗ 2 be such that Λ1 is nonsingular and define X+ 22 := U2diag(Λ−1 1 , 0)U∗ 2 (the pseudoinverse of X22); then show that Y = X12X+ 22 solves Y X22 = X12; and (c) show that X11 X12 X∗ 12 X22 = I X12X+ 22 0 I X11 − X12X+ 22X∗ 12 0 0 X22 I 0 X+ 22X∗ 12 I
Chapter 3 Linear Systems This chapter reviews some basic system theoretical concepts. The notions of controlla- bility, observability, stabilizability, and detectability are defined and various algebraic and geometric characterizations of these notions are summarized. Observer theory is then introduced. System interconnections and realizations are studied. Finally, the concepts of system poles and zeros are introduced 3.1 Descriptions of Linear Dynamical Systems Let a finite dimensional linear time invariant(FDLTI) dynamical system be described by the following linear constant coefficient differential equations Ar+ Bu, a(to)=To (3.1 Cr+ Du (3.2) There r(t)ER is called the system state, r (to) is called the initial condition of the system,u(t)ERm is called the system input, and y(t)E RP is the system output. The A, B, C, and D are appropriately dimensioned real constant matrices. A dynamical system with single-input(m= 1) and single-output (p= 1)is called a SISO(single- input and single-output) system; otherwise it is called a MIMo (multiple-input and multiple-output)system. The corresponding transfer matrix from u to y is defined as Y(s)=G(sU(s) where U(s) and Y (s)are the Laplace transforms of u(t)and y(t) with zero initial ondition (r(0)=0). Hence, we have (s)=C(s-4)-1B Note that the system equations (3. 1)and (3.2)can be a more compact matrix form B
Chapter 3 Linear Systems This chapter reviews some basic system theoretical concepts. The notions of controllability, observability, stabilizability, and detectability are defined and various algebraic and geometric characterizations of these notions are summarized. Observer theory is then introduced. System interconnections and realizations are studied. Finally, the concepts of system poles and zeros are introduced. 3.1 Descriptions of Linear Dynamical Systems Let a finite dimensional linear time invariant (FDLTI) dynamical system be described by the following linear constant coefficient differential equations: x˙ = Ax + Bu, x(t0) = x0 (3.1) y = Cx + Du, (3.2) where x(t) ∈ Rn is called the system state, x(t0) is called the initial condition of the system, u(t) ∈ Rm is called the system input, and y(t) ∈ Rp is the system output. The A,B,C, and D are appropriately dimensioned real constant matrices. A dynamical system with single-input (m = 1) and single-output (p = 1) is called a SISO (singleinput and single-output) system; otherwise it is called a MIMO (multiple-input and multiple-output) system. The corresponding transfer matrix from u to y is defined as Y (s) = G(s)U(s), where U(s) and Y (s) are the Laplace transforms of u(t) and y(t) with zero initial condition (x(0) = 0). Hence, we have G(s) = C(sI − A) −1B + D. Note that the system equations (3.1) and (3.2) can be written in a more compact matrix form: x˙ y = A B C D x u . 27
LINEAR SYSTEMS To expedite calculations involving transfer matrices, we shall use the following notation C(sI-A)B+D In MatlaB the system can also be written in the packed form using the command >>G=pck(A, B, C, D)% pack the realization in partitioned form (G)% display G in partitioned for >>A, B, C, D=unpck(G)% unpack the system matrix Note that a B C D is a real block matrix, not a transfer function. Illustrative matlab commands >>G=pck(I,I,, 10)% create a constant system matrix >>y, x, t=initial(A, B, C, D, xo)% initial response with initial condition To C, D, Iu)% impulse >>y, x]=lsim(A, B, C, D, U, T)% U is a length(T)x column(B)matrix input; T is Related MATLAB Commands: minfo, trsp, cos_tr, sin-tr, siggen 3.2 Controllability and observability We now turn to some very important concepts in linear system theory Definition 3.1 The dynamical system described by equation (3. 1)or the pair(A, B) is said to be controllable if, for any initial state r(0)=To, t1 >0 and final state T1 there exists a (piecewise continuous)input u( such that the solution of equation(3.1) satisfies r(t1)=11. Otherwise, the system or the pair(A, B)is said to be uncontrollable. The controllability (and the observability introduced next)of a system can be verified through some algebraic or geometric criteria
28 LINEAR SYSTEMS To expedite calculations involving transfer matrices, we shall use the following notation: A B C D := C(sI − A) −1B + D. In Matlab the system can also be written in the packed form using the command G=pck(A, B, C, D) % pack the realization in partitioned form seesys(G) % display G in partitioned format [A, B, C, D]=unpck(G) % unpack the system matrix Note that A B C D is a real block matrix, not a transfer function. Illustrative MATLAB Commands: G=pck([], [], [], 10) % create a constant system matrix [y, x, t]=step(A, B, C, D, Iu) % Iu=i (step response of the ith channel) [y, x, t]=initial(A, B, C, D, x0) % initial response with initial condition x0 [y, x, t]=impulse(A, B, C, D, Iu) % impulse response of the Iuth channel [y,x]=lsim(A,B,C,D,U,T) % U is a length(T ) × column(B) matrix input; T is the sampling points. Related MATLAB Commands: minfo, trsp, cos tr, sin tr, siggen 3.2 Controllability and Observability We now turn to some very important concepts in linear system theory. Definition 3.1 The dynamical system described by equation (3.1) or the pair (A,B) is said to be controllable if, for any initial state x(0) = x0, t1 > 0 and final state x1, there exists a (piecewise continuous) input u(·) such that the solution of equation (3.1) satisfies x(t1) = x1. Otherwise, the system or the pair (A,B) is said to be uncontrollable. The controllability (and the observability introduced next) of a system can be verified through some algebraic or geometric criteria