Math 477 — Homework Assignment 1, due Sept.14, 2006
1. Let Bbe a 4 ×4 matrix to which we apply the following operations:
(i) double column 1,
(ii) halve row 3,
(iii) add row 3 to row 1,
(iv) interchange columns 1 and 4,
(v) subtract row 2 from each of the other rows,
(vi) replace column 4 by column 3,
(vii) delete column 1 (so that the column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product ABC (same B) of three matrices.
2. The Pythagorean theorem asserts that for a set of northogonal vectors {xi},
n
X
i=1
xi
2
=
n
X
i=1 kxik2.
(a) Prove this in the case n= 2 by an explicit computation of kx1+x2k2.
(b) Show that this computation also establishes the general case, by induction.
3. Let A∈Cm×mbe Hermitian. An eigenvector of Ais a nonzero vector x∈Cmsuch that Ax=λx
for some λ∈C, the corresponding eigenvalue.
(a) Prove that all eigenvalues of Aare real.
(b) Prove that if xand yare eigenvectors corresponding to distinct eigenvalues, then xand y
are orthogonal.
4. What can be said about the eigenvalues of a unitary matrix?
5. Read Section 1.4 in the classnotes (Sections 2.1 and 2.2 in Kincaid/Cheney or Lecture 13 in
Trefethen/Bau contain similar information).
6. If 1
10 is correctly rounded to the normalized binary number (1.a1a2. . . a23)2×2m, what is the
roundoff error? What is the relative roundoff error?
7. Give examples of real numbers xand yfor which fl(xy)6=f l(fl(x)f l(y)). Illustrate all
four arithmetic operations using a machine with five decimal digits.
8. Consider the function f(x) = x−sin x. Since x≈sinxfor small values of x, evaluation of f
for such xinvolves a loss of significance. This loss of significance can be avoided by using the
Taylor series expansion of sin x. By using the error term of the Taylor expansion, show that at
least seven terms are required if the error is not to exceed 10−9.
9. Use Theorem 1.13 in the notes to estimate how many bits of precision are lost in a computer
when we carry out the subtraction x−sin xfor x=1
2?
