I'm working on a problem set and am trying to understand the concept of "induced norms":
Let's say I have a space of $M \times N$ matrices (I'm interpreting this as the collection of all real-valued $M \times N$ matrices). Let's say I want to convert this space into an inner-product space using some inner product $\langle A, B\rangle$. I now have some inner-product vector space where each matrix pair has an associated value produced by the inner product. For those interested, the provided inner product is $\operatorname{trace}(A^{T}B)$.
The problem then goes on to state that the provided inner product induces a norm.
What does it mean to induce a norm? And is this an induced norm on the original $M \times N$ matrix space? Or the inner-product space?
Best Answer
The norm induced by an inner product $\langle, \rangle$ is by definition $$||v|| = \sqrt{\langle v,v\rangle }.$$ You can check that the inner product properties imply that this really is a norm.