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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?

Yuriy S
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    The trace of a matrix is the sum of its eigenvalues. Here the trace is just the dot product. – smcc Jul 18 '16 at 23:38
  • @smcc, yes, I know. I'm more interested in the fact that there is only one non-zero eigenvalue – Yuriy S Jul 18 '16 at 23:39
  • But the way you have constructed it don't all the other terms cancel out because of the symmetry of the matrix. – smcc Jul 18 '16 at 23:40
  • @smcc, I see it for $2 \times 2$ matrix, but for higher dimesions? Not obvious to me – Yuriy S Jul 18 '16 at 23:41
  • The matrix has rank at most one (the column space is spanned by $\vec{a}$, if $\vec{b}$ is nonzero..). So it cannot have more than one eigenvalue. – user66081 Jul 18 '16 at 23:45
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    See: http://math.stackexchange.com/questions/55165/eigenvalues-for-the-rank-one-matrix-uvt –  Jul 18 '16 at 23:47
  • This is a well known form known as an outer product or dyadic matrix. It should not be difficult to see that its rank is 1, i.e. that the column and row spaces are both of dimension 1. That is why they have only one non-zero eigenvalue. – Tpofofn Jul 18 '16 at 23:52
  • this quesiton may help you. – Caleb Stanford Jul 18 '16 at 23:54

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