📘 Chapter 10: Diagonalization of Matrices

🔎 What is Diagonalization?

A matrix \( A \) is diagonalizable if there exists an invertible matrix \( P \) such that: \[ P^{-1} A P = D \] where \( D \) is a diagonal matrix consisting of eigenvalues of \( A \), and the columns of \( P \) are eigenvectors of \( A \).

📐 When is a Matrix Diagonalizable?

• If \( A \) has \( n \) linearly independent eigenvectors, then it is diagonalizable.
• Equivalently: Algebraic multiplicity = geometric multiplicity for each eigenvalue.

⚙️ How to Diagonalize

1. Find eigenvalues of \( A \)
2. Find a basis of eigenvectors for each eigenvalue
3. Form \( P = [\vec{v}_1 \ \vec{v}_2 \ \cdots \vec{v}_n] \)
4. Construct \( D \) with corresponding eigenvalues on the diagonal
5. Then \( P^{-1} A P = D \)

📊 Example

Let \[ A = \begin{bmatrix} 5 & 4 \\ 1 & 2 \end{bmatrix} \] - Step 1: Solve \( \det(A - \lambda I) = 0 \) - Step 2: Find eigenvectors - Step 3: Construct \( P \) and \( D \), verify \( P^{-1} A P = D \)

📝 Practice

Q: Diagonalize the matrix (if possible): \[ A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{bmatrix} \]