If A is nonsingular, show that the characteristic values of A^-1 are the reciprocals of A, and the A and A^-1 have the same characteristic vectors.

I need help getting started on this. Any help would be greatly appreciated. Thanks,

Jim

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- May 14th 2008, 04:57 AMJim NewtCharacteristic Value and Vector
If A is nonsingular, show that the characteristic values of A^-1 are the reciprocals of A, and the A and A^-1 have the same characteristic vectors.

I need help getting started on this. Any help would be greatly appreciated. Thanks,

Jim

- May 15th 2008, 04:53 AMKGene
Fix an eigenvalue $\displaystyle \lambda$ of A and let v be an eigenvector w.r.t. the eigenvalue, write down $\displaystyle Av=\lambda v$. Since A is invertible, $\displaystyle \lambda$ is nonzero and you can multiply both sides of the equation by the inverse of A.