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- - **Proof concerning eigenvalue and matrix norm.**
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Proof concerning eigenvalue and matrix norm.1 Attachment(s) Hi, I am struggling with the following proof. I think I know how to prove the right hand side inequality |lambda| <= ||A||, but I still don't know how to prove the left hand side inequality. Could you please give me some hint ? :) |

Hey hegendroffer. Hint - Use the sub-multiplicative norm with respect to the eigen-decomposition of the matrix and consider the result of the inverse of the matrix (in terms of how the eigen-values become reciprocated). The eigen-decomposition should have three matrices of PDP_inverse and the sub-multiplicative law of matrix norms in combination with how eigen-values are determined for the original matrix and the inverse relate to that result. |

Let ui and $\displaystyle \lambda_{i}$ be eigenvectors and eigenvalues of Ax=x1 u1+..xnun, ||x||=1||A||=max||Ax|| ||Ax||=||$\displaystyle \lambda_{1}$x1 u1+...$\displaystyle \lambda_{n}$xnun|| $\displaystyle \geq$ |$\displaystyle \lambda_{1}$|,..,|$\displaystyle \lambda_{n}$| ||A|| $\displaystyle \geq |\lambda|$ A u=$\displaystyle \lambda$ uA$\displaystyle ^{-1}$ u=1/$\displaystyle \lambda$ u||A$\displaystyle ^{-1}$||$\displaystyle \geq$ 1/|$\displaystyle \lambda$| |

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