On a real vector space, can all inner product be expressed by the sum of other inner product?

Question

Suppose $V$ is a real vector space, and it has inner products $\phi_1, \phi_2,...,\phi_n$. And since $\phi_1,\phi_2$ are inner products, $\phi_1+\phi_2$ can also be a new inner product in $V$. More general, just let $a,b>0,a,b\in \mathbb R$, $a\phi_i+b\phi_j$ can be a new inner product on $V$. Thus I wonder that can all inner products on a real vector space be expressed by the sum of other inner products with a positive real coefficient?

Answer 1

This is an incomplete answer (but I can't comment), hope it could help.

I suppose $V$ is an $n$-dimensional real vector space. Choosing a basis for $V$, we get a bijection between inner products and symmetric positive definite $n\times n$ matrices. My idea is that, if you can construct a basis $\{M_i\}$ for the vector space of symmetric matrices such that $M_i$ are all positive definite, then all the linear combinations of the $M_i$ with positive coefficients will give an inner product.

For example, you can take as $M_1, \dots M_n$ diagonal matrices having all diagonal entries equal to $1$ except one which equals $2$. Now complete these to a basis adding the matrices having all ones on the diagonal and zero outside except for two simmetric positions where you put $1/2$. These are all symmetric, positive definite and linearly independent. Moreover they are a basis being $n(n+1)/2$, which is the dimension of the space of symmetric matrices.

For example, in $\mathbb{R}^2$ we get $M_1=\begin{pmatrix} 2 & 0\\ 0 & 1 \end{pmatrix}$, $M_2=\begin{pmatrix} 1 & 0\\ 0 & 2 \end{pmatrix}$ and $M_3=\begin{pmatrix} 1 & 1/2\\ 1/2 & 1 \end{pmatrix}$.

Now, as you said, their linear combinations with positive coefficients of the inner products associated to $M_i$ give inner products and all the inner product (being represented by some symmetric matrix) are combination of them (but you don't know if all the coefficients are positive). It is not clear (at least to me) if all the inner product can be obtained using non negative coefficients, maybe you should take a batter basis or a bigger set.

Edit: in the $\mathbb{R}^2$ case we get $2M_1-M_2+M_3= \begin{pmatrix} 4 & 1/2\\ 1/2 & 1 \end{pmatrix}$ which is positive definite.