Let M_n(R) be the n x n matrices over the reals R. Define a norm || || on M_n(R) by ||A||= sum of absolute values of all the entries of A. Further define a new norm || ||* by ||A||* = sup{||AX||/||X||, ||X||!=0}.(adsbygoogle = window.adsbygoogle || []).push({});

Show that

1. M_n(R) under || ||* is complete.

2. If ||A||<1, then I-A is nonsingular, where I is the identity matrix.

3. The set of nonsingular matrices in M_n(R) is open.

4. Find ||B||*, where B is 2x2 and b_11=1, b_12=2, b_21=3, b_22=4.

There is a series of over 10 questions on the norm || ||. I've solved most of them but I've been stuck on (have no clue for) these ones above for a week.

I'd appreciate any hints.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Matrix Banach Space

**Physics Forums | Science Articles, Homework Help, Discussion**