How to prove that $x^\top M y = y^\top M x$ if $M$ is a symmetric matrix?

Question

I am not sure what the correct way to prove

$$x^\top M y = y^\top M x$$

is if $M$ is symmetric matrix of $k$ rows and $x$ and $y$ are vectors of length $k$. Could you give me some hint in the right direction?

Answer 1

Let use Einstein notation without $\sum$.

$($ A repeated index means sum.$)$

$$x^t*M*y=x_i(M*y)_i$$

$$=x_i.M_{ij}.y_j$$

$$=x_i.M_{ji}.y_j$$

since $M$ is symetric.

$$=y_j.M_{ji}.x_i$$

$$=y_j.(M*x)_j$$

$$=y^t*M*x$$

$i$ and $j$ are dummy indices.