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

hegendroffer April 6th, 2016 10:18 AM

Proof concerning eigenvalue and matrix norm.
 
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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 ? :)

chiro May 22nd, 2016 03:08 AM

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.

zylo June 2nd, 2016 05:30 AM

Let ui and $\displaystyle \lambda_{i}$ be eigenvectors and eigenvalues of A
x=x1u1+..xnun, ||x||=1
||A||=max||Ax||
||Ax||=||$\displaystyle \lambda_{1}$x1u1+...$\displaystyle \lambda_{n}$xnun|| $\displaystyle \geq$ |$\displaystyle \lambda_{1}$|,..,|$\displaystyle \lambda_{n}$|
||A|| $\displaystyle \geq |\lambda|$

Au=$\displaystyle \lambda$ u
A$\displaystyle ^{-1}$u=1/$\displaystyle \lambda$ u
||A$\displaystyle ^{-1}$||$\displaystyle \geq$ 1/|$\displaystyle \lambda$|


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