Hotelling Defaltion is a form of [[Deflation Methods|Deflation Method]] for lowering the [[Rank]] of a [[Matrix]]. Let $\vec v_{1}$ be a [[Eigenvalue]] of $A$ with [[Eigenvalue]] $\lambda_{1}$. Let $\vec u_{1}$ be such that $A^{\intercal}\vec u_{1} = \lambda_{1}\vec u_{1}$ and $\vec u_{1}^{\intercal}\vec v_{1}=1$.We then use $\vec u_{1}$ as our $\vec x_{1}$ value for [[Deflation Methods|deflation]]. This method is special because after deflation, the resulting [[Matrix]] $B$ will retain both all other [[Eigenvalue|Eigenvalues]] *and* [[Eigenvector|Eigenvectors]]. If $A$ is a [[Symmetric Matrix]], then $A=A^{\intercal}$. Let $\hat v_{1}$ be a [[Unit Vector|Unit]] [[Eigenvector]] with [[Eigenvalue]] $\lambda_{1}$. $ \huge \begin{align} A^{(1)} &= A \\ A^{(2)} &= A - \lambda_{1} \hat v_{1} \hat v_{1}^{\intercal} \end{align} $ Find a unit [[Eigenvector]] $\hat v_{2}$ of $A^{(2)}$ with [[Eigenvalue]] $\lambda_{2}$. $\huge \begin{align} A^{(1)} \hat v_{2} &= \pa{A^{(2)}+\lambda_{1} \hat v_{1}\hat v_{1}^{\intercal}} \hat v_{2} \\ &= A^{(2)} \vec v_{2} + \lambda_{1} \hat v_{1}\pa{\hat v_{1}^{\intercal} \hat v_{2}} \\ &= \lambda_{2} \hat v_{2} \end{align} $