University of Pennsylvania
ESE 500: Linear Systems Theory
Homework I
Due: on 9/26 at 23:59 on Gradescope
INSTRUCTIONS
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Problem 1 (15 points). Fundamental subspace decomposition.
Let U , V be finite dimensional vector spaces and A : U −→ V a linear operator. Suppose A∗ is
the adjoint operator of A and let B ∈ Rm×n.
a. (3 points) Prove the rank-nullity theorem, i.e. dim(R(B)) + dim(N(B)) = n.
b. (2 points) Prove N(A) = N(A∗A).
c. (2 points) Prove R(A)⊥ = N(A∗).
d. (2 points) Prove R(A∗) = R(A∗A).
e. (1 point) Prove R(A) = R(AA∗).
f. (3 points) Prove rank(BTB) = rank(B) = rank(BT ).
g. (2 points) Suppose the linear operator A has an inverse. Prove that the inverse is also
linear.
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Problem 2 (10 points). Suppose A1,A2 : R3 → R2, s.t. for all x = [x1, x2, x3]T ∈ R3:
A1(x) = (0.23x1 − 0.72x3,−0.1x1 − 0.4x2 + 0.3x3)
A2(x) = (x2,min (x1,−0.3x2))
A3(x) = (x1 − 5,−0.1x1 − 0.4x2 + 0.3x3)
with respect to the standard bases.
a. (3 points) Prove that A1 is a linear operator. Does that stand for A2 or A3? Explain.
b. (2 point) Find the matrix representation A1 of A1 with respect to the standard bases.
c. (2 point) Find the matrix representation of the adjoint transformation A∗1 with respect
to the standard bases.
d. (3 points) What is the matrix representation Aˆ1 of A1 with respect to the new bases
B1, B2?
B1 =
(01
1
 ,
10
1
 ,
10
0
), B2 = ([01
]
,
[
1
1
])
Problem 3 (10 points). Suppose A ∈ Rn×n. Let λ1, λ2, ..., λn ∈ C be all eigenvalues of A (they
may or may not be distinct). Also, let vi (with ‖vi‖2 = 1) for i ∈ {1, ..., n} be an eigenvector
associated with λi.
a. (2 points) Suppose AT = A. Prove ∀i ∈ {1, .., n} : λi ∈ R.
b. (2 points) Suppose AT = A. Prove that there exists a set of eigenvectors of A which is
an orthogonal basis for Rn.
c. (2 points) Suppose {v1, ..., vn} is an orthogonal basis for Rn. Prove AT = A.
d. (2 points) Prove that λ1, λ2, ..., λn are also eigenvalues of A
T .
e. (2 points) Suppose A is a positive definite matrix, that is, AT = A and for any x ∈ Rn :
xTAx > 0. Prove ∀i ∈ {1, ..., n} : λi > 0.
Problem 4 (10 points). Suppose A,B ∈ Rn×n. Let λ1, λ2, ..., λn ∈ R be eigenvalues of A.
a. (2 points) Prove
∣∣xTAx∣∣ ≤ ‖A‖2 ‖x‖22.
b. (2 points) Prove maxi |λi| ≤ ‖A‖2.
c. (2 points) Suppose AT = A and λmin = mini λi, λmax = maxi λi. Prove
∀x ∈ Rn : λmin xTx ≤ xTAx ≤ λmax xTx
d. (1 point) Consider vectors v and u. Prove that ‖v + u‖22 + ‖v − u‖22 = 2 ‖v‖22 + 2 ‖u‖22.
Problem 5 (10 points). Let C1 ∈ Rn×n, C2 ∈ Rm×m, A ∈ Rm×n ,b ∈ Rm. Suppose C1, C2
are full rank matrices and define Q1 = C
T
1 C1 and Q2 = C
T
2 C2. Consider inner products
< x, y >Q1= x
TQ1y in Rn and < x, y >Q2= xTQ2y in Rm. Also, let ‖.‖Q1 and ‖.‖Q2 be the
induced norms of these inner products on Rn and Rm, respectively. Suppose N(A) = {0}. Find
x ∈ Rn which minimizes ‖b−Ax‖Q2 in terms of A, b and Q2.
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Problem 6 (5 points). Let L : Rn×n −→ Rn×n be a linear map. We call λ ∈ C an eigenvalue
and V ∈ Cn×n an eigenmatrix of L, if L(V ) = λV . Suppose L(X) = AXAT + X where
A ∈ Rn×n. Find eigenvalues and eigenmatrices of L in terms of eigenvalues and eigenvectors
of A.
Problem 7. (Least Squares & System Identification)[8 points]
Suppose a system P as in Fig 1, which is described by some unknown dynamics and a set of
noisy data (ui, yˆi) where ui,yˆi are the input and the output of the system P at k = i, respectively.
Assume that the collected data come from a model described by the following equation
yˆk+1 = a0yk + ...+ apyk−p + b0uk + ...+ bpuk−p + wk+1 (1)
where a0, ..., ap, b0, ...bp are the unknown parameters of the system P and wk is i.i.d ∼ N (0, 1).
Given the set of data (uk, yˆk) for k = 0, 1, ...n and assuming we start the system at rest, rewrite
the previous form as
yˆk+1 = φ
T
k θ + wk+1, k = 0, 1, ..., n− 1 (2)
where φTk = [yk, ..., yk−p, uk, ..., uk−p]
T is called the feature vector and θT = [a0, ..., ap, b0, ..., bp]
T ∈
R2p+2 is the vector of the real parameters of the system, which are unknown.
System P
u y
Figure 1: Input-Output System
a) (5 points) By formulating the least squares problem
θˆn = arg min
θ
n−1∑
k=0
(yˆk+1 − φTk θ)2 (3)
compute the unique estimator θˆn of θ, given that (
n−1∑
k=0
φkφ
T
k )
−1 exists.
b) (3 points) The inputs of the systems are variables that we can control. Assume now that
uk = Kyk. What issue arises in this case? Conclude that even for this simple identification
problem, the nature of the inputs of the system is of extreme importance. This problem is
called persistent excitation in the input signal u.
Problem 8 (15 points). Let A,B ∈ Rn×n.
a. (3 points) Prove that if I −AB is invertible then I −BA is invertible. Note: We do not
know anything about invertibility of A and B.
b. (3 points) Prove that 〈X,Y 〉 = tr (X>Y ) defines an inner product, where X,Y ∈ Rn×n
and > denotes the transpose of a matrix.
c. (3 points) Suppose A ∈ Rn×n is given. Prove that there is no X ∈ Rn×n such that
AX −XA = I.
d. (2 point) Suppose A+B = AB. Show that AB = BA.
e. (2 point) Suppose A,B ∈ Rn×n are symmetric and skew-symmetric matrices respectively.
Prove that tr(AB) = 0, where tr denotes the trace of a matrix.
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f. (2 point) Suppose A,B ∈ Rn×n are symmetric matrices. Prove that if ∀x ∈ Rn : xTAx =
xTBx then A = B.
Problem 9 (7 points). Find the Jordan form representation A = QJQ−1 of the following
matrix:
A =

1/3 0 0 0
0 −1/2 1/2 0
0 1/2 −5/4 0
−1/2 0 0 1/3
 .
Note: You need to compute both Q, J . First, find the eigenvalues and eigenvectors. Then,
compute the generalized eigenvectors. Identify the Jordan blocks and state the algebraic and
geometric multiplicity of each eigenvalue. You should show your own calculations for credit.
Problem 10 (10 points). Whenever B can be derived from A by a combination of elementary
row and column operations, we write A ∼ B, and we say that A and B are equivalent matrices.
Since elementary row and column operations are left-hand and right-hand multiplication by
elementary matrices, we can say that A ∼ B ⇔ PAQ = B for nonsingular P and Q.First show
if A is m × n such that rank(A) = r, then A ∼ Nr =
(
Ir 0
0 0
)
where Nr is called the rank
normal form for A. Using above definition explain why rank(
(
A 0
0 B
)
) = rank(A) + rank(B).
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