Part 1: Proving that $A^2$ has eigenvalues $\lambda_1^2, \lambda_2^2, \dots, \lambda_n^2$

Let $A$ be an $n \times n$ matrix with eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$. This means that for each eigenvalue $\lambda_i$, there exists a non-zero eigenvector $\mathbf{v}_i$ such that: $A \mathbf{v}_i = \lambda_i \mathbf{v}_i$ Now, consider the matrix $A^2$. We want to prove that the eigenvalues of $A^2$ are $\lambda_1^2, \lambda_2^2, \dots, \lambda_n^2$.

To do this, apply the matrix $A^2$ to the eigenvector $\mathbf{v}_i$: $A^2 \mathbf{v}_i = A(A \mathbf{v}_i) = A(\lambda_i \mathbf{v}_i) = \lambda_i A \mathbf{v}_i = \lambda_i (\lambda_i \mathbf{v}_i) = \lambda_i^2 \mathbf{v}_i$ This shows that $\mathbf{v}_i$ is also an eigenvector of $A^2$, but with the eigenvalue $\lambda_i^2$.

Therefore, the eigenvalues of $A^2$ are $\lambda_1^2, \lambda_2^2, \dots, \lambda_n^2$.

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Part 2: Finding the eigenvalues of $A^{-1}$ if $A$ is non-singular

If $A$ is non-singular, it means that $A$ is invertible, and none of its eigenvalues are zero.

Let $\lambda_i$ be an eigenvalue of $A$, and let $\mathbf{v}_i$ be the corresponding eigenvector. Then: $A \mathbf{v}_i = \lambda_i \mathbf{v}_i$ Multiplying both sides by $A^{-1}$, we get: $A^{-1} A \mathbf{v}_i = A^{-1} \lambda_i \mathbf{v}_i$ Since $A^{-1} A = I$, the identity matrix, this simplifies to: $\mathbf{v}_i = \lambda_i A^{-1} \mathbf{v}_i$ Multiplying both sides by $\frac{1}{\lambda_i}$, we obtain: $A^{-1} \mathbf{v}_i = \frac{1}{\lambda_i} \mathbf{v}_i$ This shows that $\mathbf{v}_i$ is an eigenvector of $A^{-1}$ with eigenvalue $\frac{1}{\lambda_i}$.

Thus, the eigenvalues of $A^{-1}$ are $\frac{1}{\lambda_1}, \frac{1}{\lambda_2}, \dots, \frac{1}{\lambda_n}$.

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Summary:

• The eigenvalues of $A^2$ are $\lambda_1^2, \lambda_2^2, \dots, \lambda_n^2$.
• If $A$ is non-singular, the eigenvalues of $A^{-1}$ are $\frac{1}{\lambda_1}, \frac{1}{\lambda_2}, \dots, \frac{1}{\lambda_n}$.

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Would you like more details on any of these steps, or have any questions?

Here are some related questions for deeper understanding:

1. What are the conditions under which a matrix is non-singular?
2. How do eigenvalues relate to the determinant and trace of a matrix?
3. Can a matrix with a zero eigenvalue be invertible? Why or why not?
4. How does the eigenvalue-eigenvector relationship change for a diagonalizable matrix?
5. How do eigenvalues behave under matrix exponentiation?

Tip: For any matrix transformation, eigenvalues provide critical insights into how that matrix scales vectors along its eigenspaces.