How Do You Solve Norm and Matrix Approximation Problems in Linear Algebra?

[yooyo]

Q1:
how do we find a vector x so that ||A|| = ||Ax||/||x||(using the infinite norm)
totally no clue on this question..

Q2: Suppose that A is an n×n invertible matrix, and B is an approximation of A's inverse A^-1 such that AB = I + E for some matrix E. Show that the relative error in approximating A^-1 by B is bounded by ||E||(for some arbitrary matrix norm ||· ||).
relative error=(a^-1-B)/A^-1
AB=I+E--------1
AA^-1=I-------2
1/2: B/A^-1=1+E/I=1+E => (A^-1-B)/A^=-E
now I'm stucking...how do I connect -E to norm of E?
am I on the right track?any suggestion? thanks

 

[ansrivas]

1) ||Ax|| is equal to the component of y=Ax with the maximum magnitude. Now each component of y satisfies

y_i = A_i^Tx \leq \sum_j |a_{ij}|

for

||x||_{\infty} \leq 1

So the infinity norm of a matrix is

\max_{i} \sum_j |a_{ij}|

Now it is straightforward to identify the desired x.2) Your syntax does not make any sense at all: What do you mean by B/A^-1? I suggest you follow matrix convention and use the definition for relative error.