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.

 

[tacman]

For Q1, we can use the definition of the infinite norm to find a vector x that satisfies ||A|| = ||Ax||/||x||. The infinite norm of a matrix A is defined as the maximum absolute value of all entries in the matrix. So, we can find a vector x with all entries equal to 1, and then calculate ||Ax|| and ||x|| to get ||A|| = ||Ax||/||x||.

For Q2, we can use the fact that the relative error in approximating A^-1 by B is given by (A^-1-B)/A^-1. From equation 2, we know that A^-1A = I, so we can rewrite this expression as (A^-1-B)I = -E. Now, using the definition of matrix norms, we know that ||A^-1-B|| <= ||A^-1|| ||B||. Rearranging this inequality, we get ||A^-1-B||/||A^-1|| <= ||B||. Substituting this into our previous expression, we get (A^-1-B)/A^-1 <= ||B|| ||I|| = ||B||. Since ||B|| is a constant, we can say that the relative error is bounded by ||E|| for any arbitrary matrix norm ||· ||.