Numerical Analysis - Matrix Norm equalities.

[PillT]

Homework Statement

Determine constants c and C that do not depend on vector x. For this to be true:

Use this result and the definition of matrix norms to determine constants k and K so that:

Homework Equations

The Attempt at a Solution

I've done half of this problem. For c and C it was not hard. I found that if c = 1/n (n being the number of terms in the vector) and C = 1 the statement holds true.

The second half on the other hand has me stumped. The infinity norm and the one-norm only seem to share one term, due to one being the max row sum and the other being the max column sum. Can anyone give me any direction?

Thanks.

 

[KataKoniK]

To determine the constants k and K, we can use the definition of matrix norms. The infinity norm (also known as the maximum norm) of a matrix is the maximum absolute row sum, while the one-norm (also known as the column sum norm) is the maximum absolute column sum. Therefore, we can use the result from the first part of the problem to determine the constants k and K as follows:

For the infinity norm, we have k = c = 1/n and K = C = 1. This is because the maximum row sum of a matrix is equal to the maximum absolute value of any row, and since c = 1/n, the maximum absolute value of any row will be less than or equal to 1. Similarly, the maximum absolute column sum will also be less than or equal to 1, so K = 1.

For the one-norm, we have k = c = 1 and K = C = n. This is because the maximum column sum of a matrix is equal to the maximum absolute value of any column, and since c = 1, the maximum absolute value of any column will be less than or equal to n. Similarly, the maximum absolute row sum will also be less than or equal to n, so K = n.

Therefore, the constants k and K are k = 1/n, K = 1 for the infinity norm, and k = 1, K = n for the one-norm. These constants do not depend on the vector x, as required by the problem.