1. What is Matrix Diagonalization?

Matrix diagonalization is the process of finding a diagonal matrix D and an invertible matrix P such that A = PDP⁻¹. A matrix is diagonalizable if it has n linearly independent eigenvectors, where n is the dimension of the matrix.

2. How Does the Calculator Work?

The calculator finds the diagonalization using the formula:

Where:

• A — Square matrix to be diagonalized
• P — Matrix whose columns are eigenvectors of A
• D — Diagonal matrix with eigenvalues of A
• P⁻¹ — Inverse of matrix P

Explanation: The calculator first finds eigenvalues and eigenvectors, then constructs P and D matrices.

3. Importance of Diagonalization

Details: Diagonalization simplifies matrix operations, makes matrix powers easy to compute, and reveals important properties about linear transformations.

4. Using the Calculator

Tips: Enter your square matrix using comma-separated values within rows and semicolon-separated rows. For example, "1,2;3,4" represents a 2×2 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: What if my matrix isn't diagonalizable?
A: The calculator will indicate if the matrix cannot be diagonalized. Such matrices may still be put in Jordan form.

Q3: How are eigenvalues calculated?
A: Eigenvalues are found by solving the characteristic equation det(A - λI) = 0.

Q4: What's the complexity of diagonalization?
A: For an n×n matrix, eigenvalue calculation is O(n³) in general. The exact complexity depends on the algorithm used.

Q5: Can I diagonalize non-square matrices?
A: No, diagonalization is only defined for square matrices. Rectangular matrices have singular value decomposition instead.