Math 191 Wilkening
Spring 2023
Homework 2
due Sat, Feb. 4, 2:00 PM
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1. (6points)Findtherealnumbersxj intherange0
3. (4 points) Let V be a vector space with norm ∥ · ∥. Show that the reverse triangle inequality holds,
∥u − v∥ ≥ ∥u∥ − ∥v∥, u, v ∈ V .
4. (4 points) The outer product of x and y ∈ Rn is the matrix A = xyT with entries Aij = xiyj. Suppose Aˆ = f l(xyT ) and derive the componentwise forward error estimate |Aˆ − A| ≤ |A|u, where u is the unit roundoff. Why is it unlikely that Aˆ has the form (x + ∆x)(y + ∆y)T ? (As a result, computing outer products is not backward stable.)
5. (2 points) Suppose A and X are n × n real matrices and A is invertible. Show that ∥AX − I∥ ≤ κ(A)∥XA − I∥,
where I is the identity matrix and κ(A) = ∥A∥ ∥A−1∥ is the condition number of A. Here ∥ · ∥ is the operator norm associated with any vector norm on Rn.
6. (4 points) An n × n matrix L is unit lower triangular if 0 , i < j ,
Lij= 1, i=j, Lij∈R, i>j.
Show that the inverse of a unit lower triangular matrix L is also unit lower triangular.
7. (5 points) (Question 2.7, page 94 of Demmel’s book.) If A is a nonsingular symmetric matrix and has the factorization A = LDM T , where L and M are unit lower triangular matrices and D is a diagonal matrix, show that L = M.
Programming Help, Add QQ: 749389476