Product of inner products

Question

Is product of innerproduct again a inner product of two vectors? For example - Is $ (< u,v >)(< x,y >) = < m,n > $? And if yes is m and n unique and how do we calculate those?

Answer 1

Yes, it's possible to find such vector, in your example the answer could by (for example):

$$m=<u,v>x, n=y$$

or

$$m=<x,y>u, n=v$$

So it isn't unique.

Answer 2

Hint: $$\langle x, y\rangle \langle w, z\rangle = \langle \langle x, y\rangle w, z\rangle $$

Answer 3

Of course the product, which is after all a scalar (call it $S$) , can be written as an inner product of two vectors, e.g., $\vec{n} = (S, 0, 0, \ldots), \vec{m} = (1, 0, 0, \ldots)$. But the decomposition of that scalar is not unique, and in fact, you can find an infinite number of correct $\vec{m} $ cevtors for any specified non-zero $\vec{n}$.