1. What is Matrix Diagonalization?

Matrix diagonalization is the process of finding a diagonal matrix D that is similar to a given square matrix A, such that \( A = P D P^{-1} \) where P is the matrix of eigenvectors and D contains the eigenvalues.

2. How Does the Calculator Work?

The calculator performs the following steps:

Where:

• \( A \) — Original square matrix
• \( P \) — Matrix whose columns are eigenvectors of A
• \( D \) — Diagonal matrix with eigenvalues of A
• \( P^{-1} \) — Inverse of matrix P

Explanation: The calculator finds eigenvalues and eigenvectors, constructs P and D, then verifies the decomposition.

3. Importance of Diagonalization

Details: Diagonalization simplifies matrix operations like exponentiation and solving differential equations. It's fundamental in quantum mechanics, vibration analysis, and many engineering applications.

4. Using the Calculator

Tips: Enter a square matrix in the format [[a,b],[c,d]]. The matrix must be diagonalizable (have n linearly independent eigenvectors for an n×n matrix).

5. Frequently Asked Questions (FAQ)

Q1: What matrices can be diagonalized?
A: A matrix is diagonalizable if it has n linearly independent eigenvectors (where n is the matrix size). All symmetric matrices are diagonalizable.

Q2: How are eigenvalues calculated?
A: Eigenvalues are found by solving the characteristic equation \( \det(A - \lambda I) = 0 \).

Q3: What if my matrix isn't diagonalizable?
A: The calculator will indicate if the matrix isn't diagonalizable. Such matrices can often be put in Jordan form instead.

Q4: How accurate are the results?
A: Results are numerically accurate but may have rounding errors for ill-conditioned matrices.

Q5: Can I diagonalize non-square matrices?
A: No, diagonalization is only defined for square matrices.