How to set positive-definite function to be equal to the length of inputs?

Question

I want to prove or rebut the following: Given a full rank $n \times n$ matrix $\mathbf{M}$, the following equality is true: $\mathbf{v}^T \mathbf{M} \mathbf{v} = \mathbf{v}^T \mathbf{v}, $ for all $ \mathbf{v} \in \mathbb{R}^n$, iff $\mathbf{M} = \mathbf{I}$, where $\mathbf{I}$ is corresponding identity matrix. I know how to prove that elements on the diagonal of M are 1, but I cannot prove that elements outside of the diagonal are zero.

Answer 1

As it turns out, we will have $v^TMv = v^Tv$ for all $v \in \Bbb R^n$ if and only if $M + M^T = 2I$. One non-symmetric, full-rank example would be $$ M = \pmatrix{1&-1\\1&1}. $$ While the off-diagonal entries are not necessarily zero, they do satisfy $a_{ij} = -a_{ji}$.