Do matrix-orthogonal vectors have to be eigenvectors of the matrix?

Question

Say there is a square symmetric matrix $\mathbf{M}$ and vector $\mathbf{v}$. Then the vector $\mathbf{\tilde{v}}$ satisfying

$$\mathbf{\tilde{v}' Mv}=0$$

is said to be $\mathbf{M}$-orthogonal to $\mathbf{v}$ (for eg. page 48 of "Portfolio Theory" by Giorgio P. Szego, 1980).

Noting that the eigenvectors of $\mathbf{M}$ also satisfy this relation, i.e.:

$$\mathbf{e}_i' \mathbf{Me}_{j\neq i}=0$$

May I conclude that $\mathbf{v}$ and $\mathbf{\tilde{v}}$ are eigenvectors of $\mathbf{M}$?

Put another way, do there exist vectors $\mathbf{v}$ and $\mathbf{\tilde{v}}$ satisfying the first displayed equation which are not eigenvectors of $\mathbf{M}$?

Answer 1

Let $M$ be a rank 1 projection matrix, say $M=vv^t$ for some unit length $v\in\mathbb{R}^n,n>1$. So $M$ has one eigenvector, $v$. Let $\tilde{v}$ be a vector orthogonal to $v$ and then $\tilde{v}'Mv=\tilde{v}'v=0,$ although $\tilde{v}$ isn't an eigenvector.