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# Matlab數學代寫 | CS 475/675 Spring 2021: Assignment 3

### Matlab數學代寫 | CS 475/675 Spring 2021: Assignment 3

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

(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

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 modi cation 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.

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

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

(c) (6 marks) Explain how the 2  2 Householder transformation can be used in an ecient
algorithm to compute the QR factorization of a tridiagonal matrix. (Similar to the more

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

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. #### 在線客服  Essay_Cheery  