- **Abstract Algebra**
(*http://mymathforum.com/abstract-algebra/*)

- - **Proof concerning eigenvalue and matrix norm.**
(*http://mymathforum.com/abstract-algebra/330089-proof-concerning-eigenvalue-matrix-norm.html*)

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/|$\displaystyle \lambda$| |

All times are GMT -8. The time now is 06:02 AM. |

Copyright © 2018 My Math Forum. All rights reserved.