Complete Norm on M_n(R): Questions & Hints

[nubmathie]

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}.
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.

 

[g_edgar]

Problem 1. Would completeness be easier in the other norm? Is there a relation between the norms?

Problem 2. Once you prove completeness, then you can show that the series for (I-A)^{-1} converges.

Problem 3. Use Problem 2.

 

[quasar987]

For 4., notice that by linearity, if B' denotes the linear map associated to the matrix B, then ||B||*=sup{||B'(x)||: ||x||=1}. And R² can be identified (i.e. is isomorphic) as a normed linear space to C (via the correspondance (x,y)<-->x+iy). Under this correspondence, then, the map B' becomes the map C(x+iy)=(x + 2y) + i(3x + 4y) and so ||B||*==sup{||C(z)||: ||z||=1}. I have transported the problem from R² to C simply because in C, the unit vector are easily parametrized: they are just the e^it for t in [0,2pi]. And so the problem of finding ||B||* has been reduced to the simple calculus problem of finding max{||C(cos(t)+isin(t))||: t in [0,2pi]}.

 

[nubmathie]

Would you know how to show that ||AB||*<=||A||*||B||*? This would be helpful for Problem 2.

Also, how is ||B||* equal to sup{||B'(x)||: ||x||=1}? Notice that in my definition of || ||*, X refers to an nxn matrix.