AUTUMN TRIMESTER EXAM, 2021/2022
January Offering
MATH 40550
Applied Matrix Theory
Dr T. Carroll
Assoc. Prof. E.A. Cox
Assoc. Prof. H. Sˇmigoc∗
Time Allowed: 2 hours
Instructions for Candidates
Full marks will be awarded for complete and correct answers to all five questions. Show
your work, and give justifications for your answers! Answers without derivation and
justification may not be awarded any points.
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1. (a) The matrix  6 28 321 2 0
0 1 2

has eigenvalues 10, 0, 0.
(i) Determine the dimension of, and a basis for, the eigenspace of A associated
with the eigenvalue 0. Without performing any further calculations, can
you determine if the matrix A is similar to a diagonal matrix?
(ii) Find a rank 1 matrix B ∈ R5×5 that has the nonzero eigenvalue equal
to 10, and the first diagonal element (B11) equal to the last digit of your
student number.
2. (a) Let
Q =

−1 −1 0 −√2
−1 1 √2 0
−1 1 −√2 0
1 1 0 −√2
 and Σ =

7 0 0 0
0 5 0 0
0 0 3 0
0 0 0 1
 .
(i) Determine a value for α so that αQ is an orthogonal matrix, or explain
why such α does not exist.
(ii) Determine ‖·‖2 and ‖·‖F for the following matrices: Q, Σ, QΣ, ΣQ10,
QΣQT . Justify your answers!
3. (a) Let
B =
(
3 4
−16 12
)
.
Calculate the SVD of B and find the best rank one approximation of B.
(b) Let M = U1ΣU
T
2 be the singular value decomposition of M , where Σ is defined
in Question 2, and U1 and U2 are 4 × 4 orthogonal matrices. Let M2 be the
best rank 2 approximation of M .
For each of the norms ‖M‖2, ‖M‖F , ‖M2‖2, ‖M2‖F , ‖M −M2‖2, and ‖M −M2‖F ,
and condition numbers κ2(M) and κ2(MM
T ), either compute its value or ex-
plain why it cannot be determined with the information given.
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4. Let
C =

5 8 0 3
5 0 6 5
4 3 5 4
2 5 5 4
 , E =

1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
 .
The matrix C has eigenvalues 16, 2, 2
√
2− 2,−2− 2√2.
(a) Determine right and left eigenvectors of C corresponding to the eigenvalue 16.
(b) Determine the eigenvalues of M(α) = 1
16
(αC + 4(1− α)E), where α ∈ [0, 1].
(c) Compute the limit limn→∞ (M(α))
n for all values of α ∈ [0, 1] for which the
limit exists.
5. (a) Compute the Cholesky decomposition of the matrix
N =

9 6 3 0
6 13 8 3
3 8 14 8
0 3 8 14
 .
(b) Determine the minimal t for which the matrix
S =

1 0 0 0 1
0 2 0 0
√
2
0 0 3 0
√
3
0 0 0 4 2
1
√
2
√
3 2 t

is positive semidefinite or explain why such t does not exist.
—o0o—
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