2014 Problem Sheet 8

A. Algebra 1 - Linear Algebra (8)
There are five small projects listed below. Please pick at least one of them and work
on it during the vacation. Prepare either a written account (in latex) or a 15min
presentation on your results. You may want to include examples, illustrations or
other related material.
You may collaborate with one or two other students as long as each of them has at
least one project to present that differs from your own.
I hope you enjoy these projects. Happy New Year!
2014/15 UT
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Project 1: A numerical analysis approach to integration
a. Let Pn be the vector space of real polynomials of degree n or less, and let
Rb
a < b be real numbers. Show that integration a and evaluation Ex at x
for any real x define linear functionals on Pn .
b. Prove that for distinct x0 , . . . , xn ∈ [a, b] the functionals Ex0 , . . . , Exn are
linearly independent and hence form a basis for the dual space of Pn .
c. Deduce that for fixed x0 , . . . , xn there exists real numbers c0 , . . . cn such
that for all f ∈ Pn
Z
b
a
f (t)dt = c0 f (x0 ) + · · · + cn fn (xn ).
d. Find the coefficients ci when n = 2, a = 0, b = 2 and xi = i.
e. Explain why the above is of interest for numerical analysts.
Project 2: The exponential map and tangent vectors
a. Define a norm on Mn×n (C) by ||A|| = supx∈Cn ,|x|=1 |Ax|. Show that this
sup is obtained, i.e. show that there is an x ∈ Cn with |x| = 1 such that
||A|| = |Ax|.
[Use the Heine-Borel theorem!]
b. Let A ∈ Mn C and λ be the eigenvalue of A∗ A √
with largest absolute value;
∗
here A is the adjoint of A. Prove that ||A|| = λ.
[Hint: A∗ A is diagonisable.]
c. Prove that det(exp(A)) = exp(Tr(A)) for n by n matrices A. Here Tr(A)
denotes the trace of A.
1 k
d. Show that the limit Σnk=0 k!
A for n → ∞ exists and hence that exp :
Mn×n (C) → Mn×n (C) is well-defined.
e. Show that exp τ X is orthogonal (unitary) for all τ ∈ R if and only if X is
skew-symmetric (skew-Hermitian). Note, X is skew-symmetric if X t = −X
¯ t = −X.
and skew-Hermitian if X
f. When n = 1, what is the image of the line iR under exp? Explain in what
way we may think of iR as the tangent line of the unitary group U1 at the
identity. Generalise this to U2 .
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Project 3: Polar decomposition
Let V be a finite dimensional inner-product space and let, T : V → V be a linear,
non-singular map.
a. Show that there exists a unique positive P such that P 2 = T ∗ T .
b. Define S := P T −1 and show that it is an isometry.
c. Deduce that T can be written as a product of an isometry U and a positive
linear transformation P .
d. Prove that the decomposition T = U P as a product of an isometry and a
positive linear transformation is unique.
e. (Harder.) Show that the decomposition T = U P exists even if T is not
invertible, and that P is unique.
Project 4: Ergodic theorem
The goal is to prove:
Theorem: If U is an isometry on a finite dimensional inner-product space V , and
W is the subspace of solutions x to U x = x, then the sequence
An :=
1
(1 + U + · · · + U n )
n
converges as n → ∞ to the perpendicular projection PW onto W .
a. Adopt the definition of norm for matrices from Project 2 to linear operators.
For linear operators An and A on a finite dimensional vector space the
following conditions are equivalent:
(i) kAn − Ak → 0 as n → ∞;
(ii) |An x − Ax| → 0 as n → ∞ for each fixed x;
(iii) |hAn x, yi − hAx, yi| → 0 as n → ∞ for each fixed x and y.
When one and hence all of the conditions are satisfied we say that An
converges to A. (Note that these conditions are not equivalent for infinite
dimensional vector spaces.)
b. For x ∈ W , show that An x converges to x.
c. Let N be the image of of I − U . For x ∈ N show that An x converges to
zero.
[Hint: write x = y − U y.]
d. Show that N ⊥ = W and deduce the theorem.
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Project 5: Reflections
a. Let V be a finite-dimensional real inner-product space with inner product
h·, ·i, and let v be a nonzero vector in V . Show that the set
v ⊥ = {u ∈ V : hu, vi = 0}
is an (n − 1)-dimensional subspace of V .
b. Now let Sv : V → V be the map defined by
Sv (u) = u − 2
hu, vi
v.
hv, vi
Find the eigenvalues and eigenspaces of Sv . Describe the map Sv geometrically.
c. Let Φ be a nonempty set of unit vectors in R2 such that Sv (u) ∈ Φ and
2hu, vi ∈ Z for any u, v ∈ Φ. Show that |Φ| = 2, 4 or 6. Describe the
possible sets Φ geometrically.
d. (Harder) Extend your results in part c to R3 .

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