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The provided image contains blanks related to the diagonalization process of a matrix \( A \). Here's how to fill in the missing pieces in the text:

1. **First Blank**: To diagonalize \( A \), we need to find **an invertible** matrix \( P \) and **a diagonal** matrix \( D \) such that \( A = P D P^{-1} \).
   
2. **Second Blank**: The columns of \( P \) are the **eigenvectors** of \( A \), and \( P \) is called the **eigenvector** matrix for \( A \).

3. **Third Blank**: The entries of the main diagonal of \( D \) are the **eigenvalues** of \( A \).

Thus, the completed text should read:

"Let \( A \) be an \( n \times n \) matrix. To diagonalize \( A \), we need to find an **invertible** matrix \( P \) and a **diagonal** matrix \( D \) such that \( A = P D P^{-1} \). The columns of \( P \) are the **eigenvectors** of \( A \), and \( P \) is called the **eigenvector** matrix for \( A \). The entries of the main diagonal of \( D \) are the **eigenvalues** of \( A \)."

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Here are five related questions to expand on this topic:
1. What are the conditions for a matrix to be diagonalizable?
2. How are eigenvalues computed for a matrix \( A \)?
3. What is the geometric interpretation of eigenvectors?
4. Can a matrix have more than one eigenvalue? If so, how does it affect diagonalization?
5. How does the process of diagonalization relate to solving systems of linear differential equations?

**Tip**: Diagonalization is only possible if the matrix has enough linearly independent eigenvectors. This is called being "diagonalizable."

Let A be an n x n matrix. To diagonalize A, we need to find a/an matrix P and a/an matrix D such that A = PDP^-1. The columns of P are of A, and P is called the matrix for A. The entries of the main diagonal of D are of A.

Explore the process of diagonalizing a matrix through eigenvectors and eigenvalues with detailed explanations. This advanced topic in linear algebra is crucial for students at the university level, explaining the diagonalization formula and the spectral theorem.

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