本次加拿大代写是Matlab数学的Assignment

1. (6 marks) Let v be a given non-zero column vector.

(a) (2 marks) Show algebraically that the orthogonal projection matrix P = I vvT

vT v is idempo- tent, i.e. PP = P. (This property ensures that applying such a projection matrix repeatedly

has no additional eect after the rst application.)

(b) (4 marks) One characterization of an orthogonal matrix Q is that its transpose equals its

inverse, i.e., QT = Q 1. Derive (separately) the transpose and the inverse of the Householder

matrix F = I 2vvT

vT v

; and thereby show that F is an orthogonal matrix. (Hint: Use the fact

that F is an identity matrix to which a rank-one update has been applied.)

2. (10 marks) Adapt the ideas of Householder QR-factorization to derive a method to instead

compute a factorization A = QL, where L is lower triangular and Q is orthogonal. Assume that

A is square and full-rank. Give a text description of how your algorithm works, supported by

illustrations and pseudocode. (Hint: Derive a modication of the Householder approach such

that (I 2vvT =vT v)x is zero everywhere but its last component, rather than its rst.)

3. (7 marks) Use Householder transformations to perform a QR factorization of the following matrix

by hand.

Show your work, and give the resulting factors.

4. (15 marks) Let A be a symmetric tridiagonal matrix.

(a) (4 marks) In the QR factorization of A = QR, which entries of R are in general nonzero?

Which entries of Q? Explain your answer.

(b) (5 marks) Show that the tridiagonal structure is recovered when the product RQ is formed.

(Hint: Show that (i) RQ is upper Hessenberg, and (ii) RQ is symmetric.)

(c) (6 marks) Explain how the 2 2 Householder transformation can be used in an ecient

algorithm to compute the QR factorization of a tridiagonal matrix. (Similar to the more

general algorithm we saw in Lecture 19, your method here does not need to explicitly form

Q.) Determine whether the op count complexity of your proposed algorithm will be linear,

quadratic, or cubic.

5G. [CS675 students only] (10 marks)

(a) (4 marks) Implement the QL factorization method you derived in Q2. Create a MATLAB

function:

[Q,L] = QL_Factor(A)

You may assume the input is a full-rank square matrix A. The outputs are the Q and L

factors of A. Apply your method to determine the QL factors of

Give your factors up to 3 decimal places and submit your code for QL Factor.m.