Linear Algebra help

**JCIR** · July 31st 2008, 05:38 AM

Let T e an invertible linear operator. Prove that a scalar Lamda is an eigenvalue of T. if and only if the inverse of lamda is and eigen value of T^-1.

**mr fantastic** · July 31st 2008, 04:02 PM

Originally Posted by JCIR

Let T e an invertible linear operator. Prove that a scalar Lamda is an eigenvalue of T. if and only if the inverse of lamda is and eigen value of T^-1.

Could it be this simple?:

$T^{-1} v = \frac{1}{\lambda} v \Rightarrow T \, T^{-1} v = \frac{1}{\lambda} T \, v \Rightarrow \lambda v = T \, v$ .

$T \, w = \lambda w \Rightarrow T^{-1} T \, w = T^{-1}\lambda \, w \Rightarrow \frac{1}{\lambda} \, w = T^{-1} w$ .

Details and justifications left to you.

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