📂Matrix Algebra

Similar Matrices Have the Same Eigenvalues

Theorem

If two matrices $A,B$ are similar, they have the same eigenvalues.

$$\det (A - \lambda I) = \det (B - \lambda I)$$

In this case, $\lambda$ is an eigenvalue of $A, B$.

Description

Having the same eigenvalues means that the characteristic equations are the same.

Proof

To show that the eigenvalues are the same, it is sufficient to show that the characteristic equations are the same.

$$\begin{align*} \det (A - \lambda I ) =& \det ( P^{-1} (B - \lambda I ) P ) \\ =& \det P^{-1} \det (B - \lambda I) \det P \\ =& \det P^{-1} \det P \det (B - \lambda I) \\ =& \det I \det (B - \lambda I) \\ =& \det (B - \lambda I) \end{align*}$$

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