Linear Algebra and Applications Homework # 4

ECE 269: Linear Algebra and Applications

Homework # 4

via Gradescope

1. Suggested Readings. Review Summary Slides 2, 3, video lectures and Discussion Problems.
2. Problem 1: Moore–Penrose pseudoinverse. A pseudoinverse of A ∈ Rm×n
is defined as
a matrix A+ ∈ Rn×m that satisfies
AA+A = A, A
+AA+ = A
+
and AA+ and A+A are symmetric.
(a) Show that A+ is unique.
(b) Show that (ATA)
−1AT
is the pseudoinverse and a left inverse of a full-rank tall
matrix A
(c) Show that AT
(AAT
)
−1
is the pseudoinverse and a right inverse of a full-rank fat
matrix A.
(d) Show that A−1
is the pseudoinverse of a full-rank square matrix A.
(e) Show that A is the pseudoinverse of itself for a projection matrix A.
(f) Show that (AT
)
+ = (A+)
T
.
(g) Show that (AAT
)
+ = (A+)
TA+ and (ATA)
+ = A+(A+)
T
.
(h) Show that R(A+) = R(AT
) and N (A+) = N (AT
).
(i) Show that P = AA+ and Q = A+A are projection matrices.
(j) Show that y = Px and z = Qx are the projections of x onto R(A) and R(AT
), respectively, where P and Q are defined as previously.
(k) Show that x
∗ = A+b is a least-squares solution to the linear equation Ax = b, i.e.,
kAx∗ − bk ≤ kAx − bk for every other x.
∗For more information on Academic Integrity Policies at UCSD, please visit http://academicintegrity.ucsd.
edu/excel-integrity/define-cheating/index.html
(l) Show that x
∗ = A+b is the least-norm solution to the linear equation Ax = b, i.e.,
kx
∗k ≤ kxk for every other solution x, (assuming that a solution exists).
3. Problem 2: Eigenvalues. Suppose that A has λ1, λ2, . . . , λn as its eigenvalues.
(a) Show that det(A) = λ1 · λ2 · · · λn.
(b) Show that the eigenvalues of AT are λ1, λ2, . . . , λn, that is, A and AT have the same
set of eigenvalues.
(c) Show that the eigenvalues of Ak are λ
k
1
, λ
k
2
, . . . , λ
k
n
for k = 1, 2, . . . .
(d) Show that A is invertible if and only if it does not have a zero eigenvalue.
(e) Suppose that A is invertible. Show that the eigenvalues of A−1 are λ
−1
1
, λ
−1
2
, . . . , λ
−1
n
.
(f) Show that A and T
−1AT have the same set of eigenvalues, that is, eigenvalues are
invariant under a similarity transformation A 7→ T
−1AT.
4. Problem 3: Trace. We define the trace of A ∈ Rn×n as
tr(A) = A11 + A22 + · · · + Ann,
that is, the sum of its diagonal entries.
(a) Suppose that A has λ1, λ2, . . . , λn as its eigenvalues. Show that
tr(A) = λ1 + λ2 + · · · + λn.
(b) Show that
tr(A
k
) = λ
k
1 + λ
k
2 + · · · + λ
k
n
for k = 1, 2, . . . .
5. Problem 4: More on Eigenvalues. Let λ1, λ2, · · · , λn be the eigenvalues of A ∈ Cn×n
,
repeated according to their algebraic multiplicities. Show that
n
∑
i=1
|λi
|
2 ≤
n
∑
i=1
n
∑
j=1
|Ai,j
|
2
6. Problem 5: Limit. Let
A =


5 −8/5
12 −19/5


compute limn→∞
An
.