Properties of eigenvalues under matrix operations:
- If $\lambda$ is an eigenvalue of $A$, then $\lambda^k$ is an eigenvalue of $A^k$
- If $\lambda$ is an eigenvalue of $A$, then $1/\lambda$ is an eigenvalue of $A^{-1}$ (if $A$ invertible)
- If $\lambda$ is an eigenvalue of $A$, then $\lambda + c$ is an eigenvalue of $A + cI$
- If $\lambda$ is an eigenvalue of $A$, then $c\lambda$ is an eigenvalue of $cA$
- Similar matrices have the same eigenvalues
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- The eigenvalues of a scalar multiple of a matrix are the scalar multiples of the eigenvalues.If $Av=\lambda v$ with $v\ne0$, then $(cA)v=(c\lambda)v$. For $c\ne0$, all eigenvalues of $cA$ arise this way. For...
- The eigenvalues of a power of a matrix are the power the eigenvalues.If $Av=\lambda v$, then $A^kv=\lambda^kv$ for each positive integer $k$, by induction. Over $\mathbb C$, all...
- The eigenvalues of a polynomial of a matrix are the polynomial of the eigenvalues.For a polynomial $p$, $Av=\lambda v$ implies $p(A)v=p(\lambda)v$. This follows by applying the power rule term by...
- The eigenvalues of the inverse of a nonsingular matrix are the reciprocals of the eigenvalues.If $A$ is invertible and $Av=\lambda v$, then $\lambda\ne0$ and $A^{-1}v=\lambda^{-1}v$. Multiply the eigenvector...
- A matrix and its transpose have the same eigenvalues/characteristic polynomial.A square matrix and its transpose have the same characteristic polynomial because...
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