A [[Deflation Methods]] is a method for reducing the [[Rank]] of a [[Matrix]]. This is done by converting a matrix $A$ with an eigenvalue $\lambda_{1}$ into a new matrix $B$ with the same eigenvalues and [[Eigenvector|Eigenvectors]] as $A$ without $\lambda_1$. Let $A\in M_{n\times n}$ be a [[Matrix]] with [[Eigenvalue|Eigenvalues]] $\lambda_{1},\lambda_{2},\dots$ and [[Eigenvector|Eigenvectors]] $\vec \lambda_{1},\vec \lambda_{2}, \dots$ - Let $\vec x \in \R^{n}$ be such that $\vec x^{\intercal}\vec \lambda_{1}=1$ $\huge \begin{align} \vec x^{\intercal}\vec \lambda_{1} &= 1 \\ \let B &= A - \lambda_{1} \vec \lambda_{1} \vec x^{\intercal} \\ B\vec \lambda_{1} &= A\vec \lambda_{1} - \lambda_{1} \vec \lambda_{1} \vec x^{\intercal} \vec \lambda_{1} \\ &= \lambda_{1} \vec \lambda_{1} - \lambda_{1} \vec \lambda_{1} \\ & = \vec 0 \end{align} $ Let $\vec w_{1}, \dots$ be [[Eigenvector|Eigenvectors]] of $B$. $\huge \begin{align} \let \vec w_{1} &= \vec \lambda_{1} \\ \end{align} $ Let $0,\mu_{1},\dots$ be [[Eigenvalue|Eigenvalues]] of $B$. ... If $B=A - \lambda_{1}\vec \lambda_{1}\vec x^{\intercal}$ has [[Eigenvalue|Eigenvalues]] $0,\lambda_{2},\dots$ has eigenvectors $\set{\vec w_{i}}$, then $A$ has [[Eigenvalue|Eigenvalues]] $\lambda_{1},\lambda_{2},\dots$ and [[Eigenvector|Eigenvectors]] $\set{(\lambda_{i}-\lambda_{1})\vec w_{i}+ \lambda_{i}\pa{\vec x^{\intercal}\vec w_{i}}\vec \lambda_{1}}$.