MATH4141/MATH6651/PHYS5070

FW02

Assignment 4 Due date: Friday, Nov. 8

1. Show that the following matrices are orthogonal:

(a) the elementary reflectorwhere;

(b) the rotation matrix P which differs from the identity matrix in four elements, viz,:

p_ii = p_jj = c

p_ij = -p_ji = s

for i not equal to j and c² + s² = 1.

2. The Jacobi method uses a rotation matrices of the form given in question 1(b) to carry out similarity transformations of a symmetric matrix A, i.e. a sequence of matrices are generated where A⁽ⁱ⁺¹⁾ = P^TA⁽ⁱ⁾P and A⁽¹⁾ = A.

(a) If A´ = P^TAP show that the elements of A´ are identical to those of A except for

a´_ki = ca_ki - sa_kj = a´_ik

a´_kj = ca_kj + sa_ki = a´_jk

for k not equal to i or j and

a´_ii = c²a_ii + s²a_jj - 2csa_ij

a´_jj = c²a_jj + s²a_ii + 2csa_ij

a´_ij = (c² - s²)a_ij + cs(a_ii - a_jj) = a´_ji

where a_mn are the elements of A. Note that A´ is symmetric if A is.

(b) The basis of the Jacobi method is to choose c (and s) such that a´_ij = 0. If S is the sum of the off diagonal elements of A, i.e.

this shows that repeated Jacobi transformations for a sequence of (i, j) values will result in a sequence of matrices whose off diagonal elements tend to zero. The diagonal elements will then tend to the eigenvalues of A.

(c) In order to have a´_ij = 0 show that c = [1 + t²]^-1/2, s = t c and t can be chosen to be

3. The following IMSL routines implement the Housholder plus QR method to find all the eigenvalues and eigenvectors of a real, symmetric matrix:

i) imsl_d_eig_sym (C version);

ii) DEVCSF (double precision Fortran version).

Write a program to use one of these routines to find all the eigenvalues and eigenvectors of the following matrices:

(a) ;

(b) ;

(c) .

Format your output to display each of the eigenvalues with the corresponding eigenvector.

4. If a matrix A has pseudoinverse A⁺ prove it has the following properties:

i) AA⁺A = A

ii) A⁺AA⁺ = A⁺

iii) (AA⁺)^T = AA⁺

iv) (A⁺A)^T = A⁺A

Hint: If the singular value decomposition of A is UWV^T show first that W has these four properties. These properties are known as the Penrose properties.

5. The amount of a product p of a chemical reaction present at time t is measured. The results are given in the following table:

We want to find a linear least-squares fit to these data of the form ct + d = p where c and d are unknowns to be determined. If we put the data in the form of a system of linear equations

Ax = p where A is a 5 by 2 matrix whose first column consists of the values of t and the second column has all elements equal to one, x is the vector of unknowns [c, d]^T and p is the vector of p values then the least-squares solution is given by A⁺ p. Use the IMSL routines DLSGRR or lin_svd_gen to find the pseudoinverse of A. Use this to find the least squares solution to the system of linear equations given above.