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Hi could you help me with the following:

Let A be the matrix $$\pmatrix{-2 & 1& 1 \\ -2& 2& 1\\2 &-2& 3 \\ }$$

with an eigenvalue $\lambda = 2$ and corresponding eigenvector $x =[1,2,2]^T$ Construct a householder matrix $H$ ssuch that $$HAH^* = \pmatrix{ 2& *& *\\ 0 &* &*\\ 0& *& *\\}$$

I dont know how to attack the problem at all i just know the definition of Householder matrix and Householder QR factorization

Thanks a lot

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1 Answer 1

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Your displayed equation means the first column of the matrix $HAH^\ast$ is equal to $2e_1$, where $e_1=(1,0,0)^T$. That is, \begin{equation} HAH^\ast e_1=2e_1.\tag{1} \end{equation} Let $v=\frac{x}{\|x\|}$. Then $v$ is a unit eigenvector of $A$ corresponding to the eigenvalue $2$. If $v$ is the first column of $H^\ast$, then $HAH^\ast e_1=HAv=H(2v)=2Hv=2e_1$ and $(1)$ is satisfied. Hence your goal is to find a Householder matrix $H^\ast$ whose first column is $v$.

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