- #1
nubmathie
- 5
- 0
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.
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.