Is eigenvalue multiplied by constant also an eigenvalue?

Mathematics Asked by Ruby Cho on January 7, 2022

Let $A$ be an $n × n$ matrix.

If $lambda$ is an eigenvalue of $A$ and $c$ is a nonzero scalar, then $clambda$ is another eigenvalue of $A$.

I found this on "Linear Algebra and its applications (Jim Defranza)", summary of Chapter 5.

It is acceptable, that eigenvectors multiplied by constant is ok, cause $A(cv) = cAv = clambda v = lambda(cv)$.

But I don’t understand $clambda$ is also an eigenvalue of $A$.

Thank you.

One Answer

Great question, and good catch!

It's a typo/brain-o. The author surely meant to write "If $v$ is an eigenvector of $A$ and ..."

Answered by John Hughes on January 7, 2022

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