Mathematics 372 – Numerical Linear Algebra Problem Set 2 – due: February 2, 2007

I. For this problem, you will work in a decimal floating point number system, where exactly 5 decimal digits are stored, and exponents lie in the range −12 ≤ e ≤ 12. All nonzero numbers will be represented in normalized form as

±.d_{1}d_{2}d_{3}d_{4}d_{5} × 10^{e},

where d_{i} ∈ {0, 1, 2, …, 9} and d_{1} ≠ 0. All computations are done with rounding to 5 digits.

A) Compute (.32113 × 10^{3}) − (.97532 × 10^{−1}) B) Compute (.63784 × 10^{4})/(.56734 × 10^{8}) C) What are the largest and smallest numbers that can be represented in normalized

form? D) For the purposes of this part, let’s say the system also contains the one non-normalized

number 0 as well as all the numbers in normalized form. How many different numbers in all are there in this system?

E) What can you say about the spacing between “consecutive” numbers in the system? Is it always the same?

II. A) Show that if A = (a_{ij}) is a lower-triangular n×n matrix, then det(A) = a_{11}a_{22} ⋯ a_{nn}.

(Hint: Use induction on n.) B) Show the same formula for det(A) if A is upper-triangular. (Hint: recall that a

determinant can be expanded along any row or column of the matrix, if you have the correct signs.)

III. Do problem 1.3.4 from the text, but changing the right hand side to

( 3 4 6 5 )^{t} .

Do the arithmetic in two different ways: A) “By hand” using exact rational number (“fraction”) arithmetic. B) “By hand (with calculator)”, but using the floating point number system from problem

I above. To make effects of this type of arithmetic more apparent, when you do a computation using several additions, multiplications, etc., group the terms and do the rounding to 5 digits and normalization after each addition, multiplication, etc. For instance if you are computing ((a ∗ b) + c)/d, you would first multiply a and b, round that result and normalize, then add c and round and normalize that result, then divide by d and round and normalize again.

C) Compare your results in parts A and B.

IV. Do problem 1.3.17a from the text, but change the right hand side to

( 1 −3 5 5 )^{t} .

Do the calculations in the same two ways as in the last problem, and compare your results again.

V. Let A be a real symmetric n×n matrix. In this problem you will show that A is positive definite if and only if all the eigenvalues of A are strictly positive. You will need to use the following version of the Spectral Theorem:

There is an invertible matrix Q with Q^{t} = Q^{−1} and

A = Q^{t}diag(λ_{1}, …, λ_{n})Q

where “diag” means the diagonal matrix with those entries on the main diagonal. (Indeed, Q^{t} here is the matrix whose columns are the orthonormal basis of R^{n} consisting of eigenvectors of A, but you should not need to use that fact.)

A) Show that if λ_{i} > 0 for all i, then A is positive definite. B) Show conversely that if A is positive definite, then λ_{i} > 0 for all i.