Given $A \in R^{m\times n}$, I have these three norm equivalence equations:
- $\|A\|_2 \le \|A\|_F \le \sqrt {n}\|A\|_2$
- $\frac {1} {\sqrt n}\|A\|_{\infty} \le \|A\|_2 \le \sqrt {m} \|A\|_{\infty}$
- $\frac {1} {\sqrt m}\|A\|_1 \le \|A\|_2 \le \sqrt {n} \|A\|_1$
I cannot use the following:
$||A||_F = \sqrt {Tr(A^TA)}$
I need to prove each of them. I do not even know where to begin on this so ANY help would be great!
The norms are the matrix norms. You can find more at http://en.wikipedia.org/wiki/Matrix_norm. Essentially the 1 norm is just adding up each element of a column and finding the largest column sum. The infinity norm is the same as 1 norm except you add up the elements in a row and find the largest row sum. The F norm (frobenius norm) is adding up the squares of each element in A and then taking the square root.