I just noticed that for any two Cartesian vectors their dot product is precisely the only non-zero eigenvalue (if such exists) of the following matrix:
$$\vec{a}=(a_1,a_2,a_3,\dots)$$
$$\vec{b}=(b_1,b_2,b_3,\dots)$$
$$A_{mn}=a_m b_n$$
For example:
$$\vec{a}=(a_1,a_2)$$
$$\vec{b}=(b_1,b_2)$$
$$A= \left[ \begin{matrix} a_1 b_1 & a_1 b_2 \\ a_2 b_1 & a_2 b_2 \end{matrix} \right]$$
$$\left| \begin{matrix} a_1 b_1-x & a_2 b_2 \\ a_2 b_1 & a_2 b_2-x \end{matrix} \right|=x^2-(a_1b_1+a_2b_2)x=0$$
$$x_1=0,~~~x_2=a_1b_1+a_2b_2=\vec{a} \cdot \vec{b}$$
The same works for vectors of any dimension. We obtain the following characteristic polynomial of $A$:
$$x^d-(a_1b_1+a_2b_2+\dots+a_db_d)x^{d-1}=0$$
Of course for two perpendicular vectors their dot product will be zero, so the characteristic polynomial will just be:
$$x^d=0$$
Is there some deeper meaning behind all this?