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# Multiplicities and eigenvalues

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Suppose A and B are similar matrices, and that &#956; is an eigenvalue of A. We know that &#956; is also an eigenvalue of B, with the same algebraic multiplicity. Suppose that g is the geometric multiplicity of &#956;, as an eigenvalue of B. Show that &#956; has geometric multiplicity g as an eigenvalue of A.

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This provides an example of working with geometric multiplicities and eigenvalues.

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Proof:
Since and are similar matrices, then we can find an invertible matrix , such that . Suppose is an eigenvalue of with geometric multiplicity ...

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