∗ = A and
∗ = B∗A∗.
- Prove Corollary 12.3.8.
- Show that if A is an n × n matrix which has an inverse then A+ = A−1.
- Using the singular value decomposition, show that for any square matrix A, it follows
that A∗A is unitarily similar to AA∗.
- Let A,B be a m × n matrices. Define an inner product on the set of m × n matrices
Show this is an inner product satisfying all the inner product axioms. Recall for M
an n × n matrix, trace
i=1nMii. The resulting norm,
F is called
the Frobenius norm and it can be used to measure the distance between two
- Let A be an m×n matrix. Show
F = ∑
jσj2 where the σj are the singular
values of A.
- If A is a general n×n matrix having possibly repeated eigenvalues, show there is a sequence
n × n matrices having distinct eigenvalues which has the property that the
ijth entry of Ak converges to the ijth entry of A for all ij. Hint: Use Schur’s
- Prove the Cayley Hamilton theorem as follows. First suppose A has a basis of
k=1n,Avk = λkvk. Let p be the characteristic polynomial.
vk = p
vk = 0. Then since is a basis, it follows
x = 0
for all x and so p = 0
. Next in the general case, use Problem 7 to obtain a
sequence of matrices whose entries converge to the entries of
A such that
Ak has n distinct eigenvalues and therefore by Theorem 6.1.7 Ak has a basis
of eigenvectors. Therefore, from the first part and for pk the characteristic
Ak, it follows pk = 0
. Now explain why and the sense in which
- Prove that Theorem 12.4.4 and Corollary 12.4.5 can be strengthened so that the condition
on the Ak is necessary as well as sufficient. Hint: Consider vectors of the form
x ∈ Fk.
- Show directly that if A is an n × n matrix and A = A∗ (A is Hermitian) then all
the eigenvalues are real and eigenvectors can be assumed to be real and that
eigenvectors associated with distinct eigenvalues are orthogonal, (their inner product is
- Let v1,
,vn be an orthonormal basis for Fn. Let Q be a matrix whose ith column is vi.
- Show that an n × n matrix Q is unitary if and only if it preserves distances. This
. This was done in the text but you should try to do it for
- Suppose and
are two orthonormal bases for
Fn and suppose Q
is an n×n matrix satisfying Qvi = wi. Then show Q is unitary. If = 1, show there is a
unitary transformation which maps
v to e1.
- Finish the proof of Theorem 12.7.5.
- Let A be a Hermitian matrix so A = A∗ and suppose all eigenvalues of A are larger than δ2.
Where here, the inner product is
- The discrete Fourier transform maps ℂn → ℂn as follows.
Show that F−1 exists and is given by the formula
Here is one way to approach this problem. Note z = Ux where
Now argue U is unitary and use this to establish the result. To show this verify each row has
length 1 and the inner product of two different rows gives 0. Now Ukj = e−i
jk and so
kj = ei
- Let f be a periodic function having period 2π. The Fourier series of f is an expression of the
and the idea is to find ck such that the above sequence converges in some way to f.
and you formally multiply both sides by e−imx and then integrate from 0 to 2π,
interchanging the integral with the sum without any concern for whether this makes sense,
show it is reasonable from this to expect
Now suppose you only know f at equally spaced points 2
πj∕n for j = 0,1,
Consider the Riemann sum for this integral obtained from using the left endpoint
of the subintervals determined from the partition
j=0n. How does this
compare with the discrete Fourier transform? What happens as n →∞ to this
- Suppose A is a real 3 × 3 orthogonal matrix (Recall this means AAT = ATA = I. ) having
determinant 1. Show it must have an eigenvalue equal to 1. Note this shows there exists a
vector x≠0 such that Ax = x. Hint: Show first or recall that any orthogonal matrix must
preserve lengths. That is, =
- Let A be a complex m × n matrix. Using the description of the Moore Penrose inverse in
terms of the singular value decomposition, show that
where the convergence happens in the Frobenius norm. Also verify, using the singular value
decomposition, that the inverse exists in the above formula. Observe that this shows that
the Moore Penrose inverse is unique.
- Show that A+ =
+A∗. Hint: You might use the description of A+ in terms of the
singular value decomposition.
- In Theorem 12.6.1. Show that every matrix which commutes with A also commutes with
A1∕k the unique nonnegative self adjoint kth root.
- Let X be a finite dimensional inner product space and let β = be an
orthonormal basis for
X. Let A ∈ℒ be self adjoint and nonnegative and let
M be its
matrix with respect to the given orthonormal basis. Show that M is nonnegative, self
adjoint also. Use this to show that A has a unique nonnegative self adjoint kth
- Let A be a complex m×n matrix having singular value decomposition U∗AV =
as explained above, where
σ is k × k. Show that
the last n − k columns of V .
- The principal submatrices of an n × n matrix A are Ak where Ak consists those entries
which are in the first k rows and first k columns of A. Suppose A is a real symmetric matrix
and that x → is positive definite. This means that if
Show that each of the principal submatrices are positive definite. Hint: Consider
x consists of k entries.
- ↑Show that if A is a symmetric positive definite n × n real matrix, then A has an LU
factorization with the property that each entry on the main diagonal in U is
positive. Hint: This is pretty clear if A is 1×1. Assume true for
Then as above, Â is positive definite. Thus it has an LU factorization with all positive entries
on the diagonal of U. Notice that, using block multiplication,
Now consider that matrix on the right. Argue that it is of the form
Ũ where Ũ has all
positive diagonal entries except possibly for the one in the nth row and nth column.
Now explain why det
> 0 and argue that in fact all diagonal entries of Ũ are
- ↑Let A be a real symmetric n × n matrix and A = LU where L has all ones down the
diagonal and U has all positive entries down the main diagonal. Show that A = LDH where
L is lower triangular and H is upper triangular, each having all ones down the diagonal and
D a diagonal matrix having all positive entries down the main diagonal. In fact, these are
the diagonal entries of U.
- ↑Show that if L,L1 are lower triangular with ones down the main diagonal and H,H1 are
upper triangular with all ones down the main diagonal and D, D1 are diagonal matrices
having all positive diagonal entries, and if LDH = L1D1H1, then L = L1,H = H1,D = D1.
Hint: Explain why D1−1L1−1LD = H1H−1. Then explain why the right side is upper
triangular and the left side is lower triangular. Conclude these are both diagonal matrices.
However, there are all ones down the diagonal in the expression on the right.
Hence H = H1. Do something similar to conclude that L = L1 and then that
D = D1.
- ↑Show that if A is a symmetric real matrix such that x → is positive definite, then
there exists a lower triangular matrix
L having all positive entries down the diagonal such
that A = LLT. Hint: From the above, A = LDH where L,H are respectively lower and
upper triangular having all ones down the diagonal and D is a diagonal matrix having all
positive entries. Then argue from the above problem and symmetry of A that H = LT. Now
modify L by making it equal to LD1∕2. This is called the Cholesky factorization.
- Given F ∈ℒ where
X,Y are inner product spaces and dim =
n ≤ m = dim
there exists R,U such that U is nonnegative and Hermitian and R∗R = I such that F = RU.
Show that U is actually unique and that R is determined on U