Proof that norm of submatrix must be less than norm of matrix it's embedded in

[Simfish]

Homework Statement

http://dl.dropbox.com/u/4027565/2010-10-10_194728.png

Homework Equations

The Attempt at a Solution

||B|| = ||M_1 * A * M_2 ||

So from an equality following from the norm, we can get...

||B|| <= ||M_1||*||A||*||M_2||.

Now, we know that B is a submatrix of A. So if A is 4x3, then M_1 must be 1x4 and M_2 must be 3X1 (I know that block matrices are more complicated than that, but this might work). What this also means is that the combined product of M_1 and M_2 must be <= 1. But beyond that, I'm stuck. Is there another step I should take?

Thanks!

 

[Simfish]

Okay so if B is ultimately 1x1, then M_1 and M_2 must both be matrices with 0s everywhere except for one row (or column). So the maximum norm (under any situation) would be 1.

 

[arkajad]

You cane always embed B in 4x4 block matrices by adding 0 and I as appropriate blocks. Block matrices M_1 and M_2 will probably be orthogonal projections of norm ||M||=1. In any case the key is to find explicitly M_1 and M_2 and their norms. Then use $$||ABC||\leq ||A||\,||B||\,||C||.$$