----- > [!proposition] Proposition. ([[eigenvalues of outer product]]) > Let $x,y \in \mathbb{F}^{N}$. The [[matrix]] $xy'$ has a *single* possibly nonzero [[eigenvalue]] equal to $y'x$. All other [[eigenvalue]]s are $0$. > \ > As a corollary, the [[eigenvalue]]s of $xy'$ are all equal to $0$ whenever $x$ and $y$ are [[orthogonal]]. > [!proposition] Lemma 1. > $x,y \in \mathbb{F}^{N}$, the [[characteristic polynomial]] $\det(I-xy')$ is equal to $1-y'x$. \ **Proof of Lemma 1.** We have $\begin{align} \det (I-xy')= & (\det 1)\det(I-x (1)^{-1} y') \\ = & \det \begin{bmatrix} 1 & y' \\ x & I \end{bmatrix} \ \ (*) \\ = & (\det 1) \det(1-y' I^{-1}x) \ \ (**) \\ = & \det(1-y'x) \\ = & 1-y'x \end{align}$ where in $*$ and $**$ we apply the two det-matrix-inversion lemmas and the final equality follows as the [[determinant]] of a scalar. > [!proposition] Lemma 2. > As a corollary of **lemma 1**, we get that for $\lambda \neq 0$: $\begin{align} \det (\lambda I_{N} - xy')=&\det \big( (\lambda I_{N})\left( I_{N}-\frac{xy'}{\lambda} \right) \big) \\ =& \det(\lambda I_{N}) \det (I_{N}- (\frac{1}{\lambda}x)y') \\ = & \lambda^{N} (1-y'(\frac{1}{\lambda}x)) \\ = & \lambda^{N} - \lambda^{N-1}y'x, \end{align}$ and for $\lambda=0$: $\begin{align} \det (\lambda I_{N}-xy')= & \det(-xy') \\ = & (-1)^{N} \det (xy') \\ = & \begin{cases} 0 & \text{ if } N > 1 \\ -xy & \text{ if } N=1, \end{cases} \end{align}$ \ where the last equality follows because $xy'$ has [[rank]] $1$ as an [[outer product]]. > [!proof]- Proof. ([[eigenvalues of outer product]]) We know that the [[eigenvalue]]s of the [[matrix]] $xy'$ may be computed via the [[characteristic polynomial]] $\det (xy' - \lambda I_{N})=0, $ which is $0$ when and only when $\det (\lambda I_{N}-xy')=0.$ Using **lemma 2** we see that the solutions to this polynomial in lambda are the solutions to $\lambda^{N} - \lambda^{N-1}y'x=0.$ Rearranging yields $\lambda^{N}=\lambda^{N-1}y'x,$ canceling then gives us $\lambda=y'x$ as one solution and $\lambda=0$ as the other $N-1$ solutions. This is what we wanted to show. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```