- #1
pyroknife
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Homework Statement
The weighted vector norm is defined as
##||x||_W = ||Wx||##.
W is an invertible matrix.
The induced weighted matrix norm is induced by the above vector norm and is written as:
##||A||_W = sup_{x\neq 0} \frac{||Ax||_W}{||x||_W}##
A is a matrix.
Need to show ##||A||_W = ||WAW^{-1}||##
Homework Equations
The Attempt at a Solution
##||A||_W = sup_{x\neq 0} \frac{||Ax||_W}{||x||_W} = sup_{x\neq 0} \frac{||WAx||}{||Wx||} = sup_{x\neq 0} \frac{||WAW^{-1}Wx||}{||Wx||}##
For an induced norm we know that:
##\frac{||WAW^{-1}Wx||}{||Wx||} \leq \frac{||WAW^{-1}||||Wx||}{||Wx||} = ||WAW^{-1}||##Here is where I am lost. I already have gotten the expression into the form desired, but I do not know how to make the connection between ##sup_{x\neq0}## and the ##\leq##. My thought is that we are taking the supremum of ##\frac{||Ax||_W}{||x||_W}##, so it's maximum possible value is ##||WAW^{-1}||##
and thus
##||A||_W=||WAW^{-1}||##
Is this the right logic?