Hi.

Problem:Prove that if $\displaystyle W$ is an arbitrary nonsingular matrix, the function $\displaystyle ||\cdot||_W$ defined by (3.3) is a vector norm.

(3.3) $\displaystyle ||x||_W = ||Wx||$.

Attempt:I know that in if a function is a norm it has to have three properties.

(1) $\displaystyle ||x||=0 \;\textrm{if and only if}\; x=0$,

(2) $\displaystyle ||x||+||y||=||x+y||$ and

(3) $\displaystyle ||\alpha x|| = \alpha ||x||$

(1)

Let $\displaystyle x=0$.

Since $\displaystyle ||x||_W=||Wx||=||\sum^n_{i=1}w_ix_i||$ and every $\displaystyle x_i=0$, we have that $\displaystyle ||x||_W=0$.

Let $\displaystyle ||x||_W=0$.

Since $\displaystyle ||x||_W=||\sum^n_{i=1}w_ix_i||$ and $\displaystyle W$ is nonsingular such that $\displaystyle w_i\neq 0$ for $\displaystyle 1\leq i\leq n$, we have that $\displaystyle x_i=0$ and so $\displaystyle x=0$.

(3)

$\displaystyle

\begin{aligned}

|\alpha| ||x||_W=&\;|\alpha| ||Wx||\\

=&\;|\alpha| ||\sum^n_{i=1}w_ix_i||\\

=&\; |\alpha| \left(\sum^n_{i=1}(w_ix_i)^p\right)^{1/p}\\

=&\; \left( \sum^n_{i=1}(\alpha w_i x_i)^p\right)^{1/p}\\

=&\; ||\alpha x||_W

\end{aligned}

$

(2)

I do not know how to prove that the Triangle Inequality holds. Hints are greatly appreciated.

Thanks.