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###### 日期：2020-09-24 10:46

HW 2 Matlab (10pts)

What is being calculated? What do the results show? Use different choices for the error to

assist with your explanation. It is not sufficient to run the code and give the results without

the code and run. For the full 10 pts you must explain mathematically what is going on.

Explain (mathematics not the coding) for the attached code. Experiment with different

choices of the error. (10 points)

HW 2 Theoretical Questions: Norms and Eigenvalues (20pts)

1. Let A of size m × m be complex and Hermitian. An eigenvector of A is a nonzero

vector x ∈ Cm such that Ax = λx for some λ ∈ C, the corresponding eigenvalue.

(a) Prove that all eigenvalues of A are real

(b) Prove that if x and y are distinct eigenvectors corresponding to distinct eigenvalues,

then x and y are orthogonal.

2. Let S be complex of size m × m be skew Hermitian S

? = ?S.

(a) Prove that the eigenvalues of S are pure imaginary. (ie the eigenvalues are all of

the form λ = ?x for some x and Real(λ) = 0

(b) Prove that I ? S is nonsingular.

(c) Prove that the matrix Q = (I ? S)

?1

(I + S) is unitary. This is the Cayley

transform of S.

3. It u and v are vectors of length m, the matrix A = I + uv?

is known as a rank one

perturbation of the identity.

(a) Prove that if A is nonsingular, then its inverse has the form A?1 = I + αuv?

for

some scalar α. (i.e. verify that this is the inverse).

(b) What is α? Show how you found α.

(c) For what u and v is A singular? Prove your statement.

(d) If it is singular, what is null(A)? Why? Explain.

1

Basic Background on Norms.

Prove each of these statements

1. For the inner product defined by

< x, y >=Xixiyˉi

(a) < x, y > =< y, x >.

(b) < x, y + z >=< x, y > + < x, z >

(c) The Cauchy Schwartz inequality

| < x, y > | ≤ √

< x, x >< y, y >.

2. The inequalities which relate the 1, 2 and ∞ norms.

(a) ||x||2 ≤ ||x||1 ≤

n||x||2 You will need to use Holder’s inequality for complex

numbers ai

, where λmax denotes the largest eigenvalue of the given

matrix.

4. The inequalities which relate the 1, 2 and ∞ matrix norms for square matrices of

dimension n.