Math 423 Homework -- Spring 2002

Assignment 1.

• Read sections 2.5, 2.6, and 3.5
• Exercises: § 2.5 - 4,6; § 2.6 - 2, 10; § 3.5 - 3, 6

Assignment 2.

• Read sections 5.1 to 5.6.
• Exercises: § 3.7 - 5, 9(c); § 5.3 - 4, 6, 19, 20(a); § 5.4 - 8, 12, 14

Assignment 3.

• Read sections 5.6 to 5.8.
• Exercises: § 5.4 - 16(a,b), 17, 25; § 5.5 - 1;

Assignment 4.

• Read sections 5.8, 5.9, 6.1.
• Exercises: § 5.5 - 8, 11, 12; § 5.6 - 2, 8(b), 9; § 5.7 - 2, 5, 12, 16

Assignment 5.

• Read sections 6.1-6.3
• Exercises: § 5.8 - 11, 12, 15, 16; § 5.9 - 2, 7, 20, 23

Assignment 6.

• Read sections 6.3-6.4
• Exercises: § 6.1 - 5, 6, 16, 22; § 6.2 - 3, 4

Assignment 7.

• Read sections 6.4, 7.1-7.2
• Exercises: § 6.2 - 7, 9, 13, 14; § 6.3 - 4, 10, 11

Assignment 8.

• Read sections 7.3-7.5
• Exercises: §6.4 - 6, 11, 12, 13; §7.1 - 5, 6, 8; §7.2 - 2.

Assignment 9.

• Read sections 7.3-7.5
• Exercises: §7.2 - 14, 17, 18; §7.3 - 2, 6(d), 10(e); §7.4 - 1, 5(a), 11, 13.

• Read the class notes.
  1. Consider the matrix:

     B	=	1	1
     		-1	1
     		1	2

     1. Find the SVD for B.
     2. Use the SVD for B to find the SVD for B^T. How do the singular values for the two compare?
     3. Use this observation to show that the eigenvalues of B^TB and BB^T are the same, except for 0.
  2. Consider the matrix:

     B	=	1	-1
     		1	-1.001

     1. Find the SVD for B.
     2. For a square matrix, the ratio of the largest to smallest singular values is called the condition number of the matrix. Find the condition number for B.
     3. Suppose that û is the measure or calculated estimate of a vector u. The relative error is then
        r_err(u) = || û - u ||· || u ||^-1.
        Solve the systems Bu = [1 1]^T and Bû = [0.999 1.001]^T exactly. Calculate r_err(u). How does this compare with the relative error in the right hand side, which is 0.001? How does the ratio of r_err(u) to 0.001 compare with the condition number for this problem? Briefly explain your results.
  3. Let C be an n×n self-adjoint matrix; that is C=C^H. Show that all of the eigenvalues of C are positive if and only if for all nonzero vectors X we have X^HC X > 0. Such matrices are called positive definite.
  4. Verify that if C is a positive definite matrix, then
     (X,Y) := Y^HC X
     defines an inner product on Cⁿ.