Is 1/λ an Eigenvalue of the Inverse of A?

Thread starter stine23
Start date Dec 3, 2008
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Eigenvalue Matrices

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SUMMARY

If A is an invertible matrix and λ is a non-zero eigenvalue, then 1/λ is definitively an eigenvalue of the inverse matrix A⁻¹. This conclusion arises from the relationship Av = λv, where v is an eigenvector. By manipulating this equation, we find that A⁻¹(λv) = v, which leads to the identification of 1/λ as an eigenvalue of A⁻¹.

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Dec 3, 2008

stine23

How do I prove that if A is an invertible matrix and lambda does not equal zero then one dived by lambda is an eigenvalue of the inverse of A?

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Dec 3, 2008

Ben Niehoff

Science Advisor

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First of all, if A is invertible, then none of the eigenvalues can be 0, by definition.

Second, consider

Av = \lambda v

for some eigenvector v. What is

A^{-1}(\lambda v)

?