MATH4141/MATH6651/PHYS5070

FW02

Assignment 3 Due date: Monday, Oct. 28

1. If A is an n by n matrix with an eigenvalue s and corresponding eigenvector x show that the matrix (A - qI)^-1 has an eigenvalue (s - q)^-1 with the same corresponding eigenvector x provided q is not equal to one of the eigenvalues of A.

2. Let A be an n by n matrix with an eigenvalues s₁, s₂, ... s_n and corresponding eigenvectors v⁽¹⁾, v⁽²⁾, ..., v⁽ⁿ⁾. If x is any vector with the property that x^Tv⁽¹⁾ = 1 show that the matrix B = A - s₁v⁽¹⁾x^T has eigenvalues 0, s₂, ... s_n with corresponding eigenvectors v⁽¹⁾, w⁽²⁾, ..., w⁽ⁿ⁾ where w⁽ⁱ⁾ is defined via the relation

v⁽ⁱ⁾ = (s_i - s₁)w⁽ⁱ⁾ + s₁(x^Tw⁽ⁱ⁾ ) v⁽¹⁾

for i = 2, ..., n.

3. Use the Gerschgorin circle theorem to estimate the location of the eigenvalues of the following matrices. Draw a rough diagram to illustrate your results.

(a) (b)

4. Use mathematical induction to prove the formulae for y_k, 1/p_k² and μ_k given in the notes on the symmetric power method.

5. Write a program to implement the symmetric power method for finding the dominant eigenvalue and corresponding eigenvector if a symmetric matrix. Apply it to the matrix given in part (a) of question 3. Also apply it to the matrix

Use several different choices for the starting vector x₀ in each case and compare the number of iterations required for convergence.

(N.B. If you follow Algorithm 9.2 of Burden and Faires be aware that the convergence criterion in Step 6 will not work if the dominant eigenvalue has a negative sign. Replace it by one of your own design).

6. Modify the program from question 5 to implement the inverse power method to find another eigenvalue and eigenvector for the matrices referred to there. Use an IMSL routine to solve the set of linear equations. Choose one that allows you to store the LU decomposition and use it without recalculation. Take the shift q = 9 in both cases.