MATH 520 — REVIEW FOR FINAL EXAM

B. Cole
December 11, 2012
MATH 520 — REVIEW FOR FINAL EXAM
1. Let F be the Fourier matrix for n = 3, i.e.,
⎡
⎤
1 1 1
F = ⎣ ⎦
1 w w 2
1 w 2 w 4
where w = exp((2πi)/3) = − 1
2 + i √ 3
2 , and let U = 1 √ 3 F .
(a) Calculate U 2 . Show that it is a permutation matrix.
(b) Show that U 4 = I.
(c) Calculate trace(U) and trace(U 2 ).
(d) Determine the eigenvalues for U.
⎡
(e) Show that x = ⎣ 0
⎤
1 ⎦ is an eigenvector for U. What is the corresponding eigen-
−1
value?
(f) Show that there exists a (real) orthogonal matrix Q and a diagonal matrix Λ with
F = QΛQ T .
2. (a) Let A =
1 0 i 0
. Find a basis for the row space of A. Find a basis for the
0 1 0 i
nullspace of A. Show that the row space = the nullspace.
(b) Let A be a m×n matrix (possibly complex) with row space = nullspace and
rank(A) = m. What are the possible values for n?
(c) Does there exist a non-zero real matrix A with row space = nullspace? If so, give
an example. If not, explain why.
3. Let A =
1 + i 2 + i
.
3 + i 4 + i

(a) Find L and U so that A = LU where L is lower triangular with 1’s on the diagonal
and U is upper triangular.
(b) Solve Ax = b where b =
4. Let
Find A −1 .
5. Let
1
.
i
⎡
1 0 0
⎤
0
⎢ 2
A = ⎣
3
1
4
0
1
0 ⎥
⎦ .
0
1 1 1 1
⎡
0 0 1/4
⎤
1
⎢ 1/4
A = ⎣
1/2
0
1
3/4
0
0 ⎥
⎦ .
0
1/4 0 0 0
(a) Find k with A k > 0. Conclude that A is a regular Markov matrix.
(b) Find A ∞ = limk→∞ A k .
6. (a) Let A be a non-zero 3×3 real, anti-symmetric matrix with A 2 = D, which is
diagonal. Show that 7 of the 9 entries in A are zero.
(b) Let A be a non-zero 4×4 real, anti-symmetric matrix with A 2 = D, which is
diagonal. Show that the result in (a) does not generalize. Find such an A with each
non-diagonal entry = ±1 (and furthermore A 2 = −3I).
7. Let
⎡
0
A = ⎣ −1
1
0
⎤
1
1 ⎦ .
−1 −1 0
(a) Find the eigenvalues and an orthonormal basis of eigenvector. Show that A =
UΛU H where U is unitary and Λ is diagonal.
2

(b) Note that A 2 is real symmetric. Show that A 2 = QΛ 2 Q T where Q is a (real)
orthogonal matrix.
zero.
(c) Set C = Q T AQ. Show that C = −C T and C 2 = Λ 2 . Show that 7 entries in C are
8. Let
⎡
−1
A = ⎣ 2
−2
4
⎤ ⎡
−i
2i ⎦ = ⎣
1 2 i
−1
⎤
2 ⎦ [ 1 2 i ] = ab
1
H .
Find unitary U and V and a diagonal matrix Σ with Σ ≥ 0 and A = UΣV H . (This
generalizes svd for complex matrices.)
9. Let
A =
1 1 1
.
1 1 −1
Find orthogonal matrices U, V and a diagonal matrix Σ with Σ ≥ 0 so that A = UΣV T .
10. Consider data points (t1, b1), . . . , (t5, b5) in R 2 . Find the polynomial f(t) = c0 +c1t+
c2t 2 that best fits these data points in the sense of least squares, i.e. for ei = bi − f(ti),
we minimize e 2 1 + · · · + e 2 5. Assume that
t k 1 + · · · + t k 5 = sk
where s0 = 5, s1 = 0, s2 = 4, s3 = 0, s4 = 6 and
where r0 = 1, r1 = 2, r2 = 3.
11. Let a, b, c be real numbers. Let
A =
b1t k 1 + · · · + b5t k 5 = rk
⎡ ⎤ ⎡
⎤
a 0 0 b
a 0 b
a b
, B = ⎣ 0 b 0 ⎦ ⎢ 0 b 0 0 ⎥
, C = ⎣
⎦ .
b c
0 0 b 0
b 0 c
b 0 0 c
(a) For which a, b, c is A positive definite?
3

(b) For which a, b, c is B positive definite?
(c) For which a, b, c is C positive definite?
(d) Is A positive definite exactly when B is positive definite? If not, give some choice
of a, b, c where this difference occurs.
(e) Is B positive definite exactly when C is positive definite? If not, give some choice
of a, b, c where this difference occurs.
12. Let
⎡
7 14 10 1
⎤
2
⎡
7 10 6
⎤
1
⎢ 5
A = ⎣
2
10
4
9
4
−3
−2
7 ⎥
⎦ ,
4
⎢ 5
S = ⎣
2
9
4
6
4
1 ⎥
⎦ ,
1
1 2 2 −1 2
1 2 3 1
S −1 ⎡
1 −2 3
⎤
−2
⎡
1 2 0 3
⎤
−4
⎢ −1
= ⎣
1
3
−4
−6
10
4 ⎥
⎦ ,
−7
⎢ 0
B = ⎣
0
0
0
1
0
−2
0
3 ⎥
⎦
0
−2 8 −21 16
0 0 0 0 0
where A = SB. Find a basis for each of the four fundamental subspaces for A. No
arithmetic is required here; it has already been done for you.
13. Let A be a rank 1 n×n matrix with n > 1. We know that A = uv T for certain vectors
u, v in R n .
(a) Show that u is an eigenvector for A. Express the corresponding eigenvalue in terms
of u and v.
(b) Show that A 2 = cA for some constant c.
(c) Show that A is diagonalizable if and only if c = 0.
(d) Assume c = 0. Find simple formulas for A 3 , A 4 , . . . , A k for any k ≥ 1.
(e) Assume c = 0. Find a simple formula for e tA . Show that e tA = α(t)I + β(t)A
where α(t) and β(t) are specific functions of t.
4

14. Let
B =
−1 3
.
1 −3
(a) Calculate e tB . Show that e tB is a regular Markov matrix for all positive t.
(b) Calculate limt→∞ e tB . (Note that this limit must be the same as limk→∞ A k where
A is the regular Markov matrix A = e B .)
(c) Let x(t) be a solution to x ′ (t) = Bx(t). Find a fixed vector w and a constant c so
that w · x(t) = c for all t.
15. Let
⎡
1 1 1
⎤
1
⎢ 1
A = ⎣
1
1
1
1
1
2 ⎥
⎦ .
2
1 1 1 2
(a) Find an orthonormal basis for the nullspace of A.
(b) Find a (real) orthogonal matrix Q so that B = AQ where B = [0|C] and C is a
4×2 matrix of rank 2. (You are not asked to compute B or C.)
16. Let
⎡
1 1 1
⎤
1
⎢ 1
A = ⎣
1
1
1
1
1
1 ⎥
⎦ .
1
1 1 1 1
Find a (real) orthogonal matrix Q and a diagonal matrix Λ so that A = QΛQ T .
⎡
17. Find a 3×3 (real) orthogonal matrix Q where the first column is a multiple of ⎣ 1
⎤
2 ⎦.
2
18. Let
⎡
7
A = ⎣ −4
−4
1
⎤
−4
−8 ⎦ .
−4 −8 1
Find a (real) orthogonal matrix Q and a diagonal matrix Λ so that A = QΛQ T .
5

19. Which of the matrices described below are diagonalizable? Assume that each matrix
is n×n .
(a) projection
(b) permutation
(c) real symmetric
(d) real orthogonal
(e) real anti-symmetric
(f) unitary
(g) Hermitian
(h) skew-Hermitian
(i) A + I where A is diagonalizable.
(j) A −1 where A is diagonalizable and invertible.
(k) A 2 where A is diagonalizable.
(l) I + A + A 2 where A is diagonalizable.
(m) A + B where A and B are diagonalizable.
(n) AB where A and B are diagonalizable.
(o) AB where A and B are real orthogonal.
(p) A + B where A and B are Hermitian.
(q) upper triangular.
(r) A where A 2 = A.
(s) F AF −1 where A is diagonalizable and F is invertible.
6

20. (a) Let
For which a, b, c is A diagonalizable?
(b) Let
For which a, b, c is A diagonalizable?
(c) Let
For which a, b, c is A diagonalizable?
(d) Let
⎡
1
A = ⎣ 0
a
1
⎤
b
c ⎦ .
0 0 1
⎡
1
A = ⎣ 0
a
1
⎤
b
c ⎦ .
0 0 2
⎡
1
A = ⎣ 0
a
2
⎤
b
c ⎦ .
0 0 1
⎡
1 a b
⎤
c
⎢ 0
A = ⎣
0
1
0
d
2
e ⎥
⎦ .
f
0 0 0 2
For which a, b, c, d, e, f is A diagonalizable?
(e) Let
⎡
1 a b
⎤
c
⎢ 0
A = ⎣
0
2
0
d
1
e ⎥
⎦ .
f
0 0 0 2
For which a, b, c, d, e, f is A diagonalizable?
7