∗ = A and
∗ = B∗A∗.
- Prove Corollary 14.11.10.
- 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,
i=1nMii. The resulting norm,
F is called the Frobenius norm and it can be used
to measure the distance between two matrices.
- It was shown that a matrix A is normal if and only if it is unitarily similar to a diagonal matrix. It
was also shown that if a matrix is Hermitian, then it is unitarily similar to a real diagonal matrix.
Show the converse of this last statement is also true. If a matrix is unitarily similar to a real diagonal
matrix, then it is Hermitian.
- Let A be an m×n matrix. Show
F = ∑
jσj2 where the σj are the singular values of
- If A is a general n×n matrix having possibly repeated eigenvalues, show there is a sequence of
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 theorem.
- Prove the Cayley Hamilton theorem as follows. First suppose A has a basis of eigenvectors
k=1n,Avk = λkvk. Let p be the characteristic polynomial. Show
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 8 to obtain a sequence of matrices whose entries converge to the
A such that Ak has n distinct eigenvalues and therefore by Theorem 6.5.1 on
Page 386 Ak has a basis of eigenvectors. Therefore, from the first part and for pk the
characteristic polynomial for
Ak, it follows pk = 0
. Now explain why and the sense in which
- 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 zero).
- 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 means =
This was done in the text but you should try to do it for yourself.
- 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. This is done in the text but do it yourself with all
- 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,
,n. 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 approximation?
- 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
- 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
- In Theorem 14.10.1. 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
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 root.
- 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
> 0. Show that each of the principal
submatrices are positive definite. Hint: Consider
x consists of k
- ↑A matrix A has an LU factorization if it there exists a lower triangular matrix L having all ones on
the diagonal and an upper triangular matrix U such that A = LU. 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
> 0 and argue that in fact all diagonal entries of Ũ are positive.
- ↑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
- ↑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
exists R,U such that U is nonnegative and Hermitian (U = U∗) and R∗R = I such that F = RU.
Show that U is actually unique and that R is determined on U
. This was done in the book, but
try to remember why this is so.
- If A is a complex Hermitian n × n matrix which has all eigenvalues nonnegative, show that there
exists a complex Hermitian matrix B such that BB = A.
- ↑Suppose A,B are n × n real Hermitian matrices and they both have all nonnegative
eigenvalues. Show that det
≥ det + det
Hint: Use the above problem and the
Cauchy Binet theorem. Let P2 = A,Q2 = B where P,Q are Hermitian and nonnegative.
- Suppose B = is an
× Hermitian nonnegative matrix where
α is a scalar
and A is n × n. Show that α must be real, c = b, and A = A∗,A is nonnegative, and that if α = 0,
then b = 0. Otherwise, α > 0.
- ↑If A is an n × n complex Hermitian and nonnegative matrix, show that there exists an upper
triangular matrix B such that B∗B = A. Hint: Prove this by induction. It is obviously true if n = 1.
Now if you have an
× Hermitian nonnegative matrix, then from the above problem, it
is of the form
- ↑ Suppose A is a nonnegative Hermitian matrix (all eigenvalues are nonnegative) which is partitioned
where A11,A22 are square matrices. Show that det
Hint: Use the above
problem to factor A getting
Next argue that A11 = B11∗B11,A22 = B12∗B12 + B22∗B22. Use the Cauchy Binet theorem
to argue that det = det
. Then explain why
- ↑ Prove the inequality of Hadamard. If A is a Hermitian matrix which is nonnegative (all eigenvalues
are nonnegative), then det