Matrix Diagonalization Calculator
Decompose matrix A into A = PDP⁻¹ using Eigenvalues and Eigenvectors.
Matrix A (2×2)
Eigenvalues (λ)
Matrix P (Eigenvectors)
Matrix D (Diagonal)
Eigenspace Visualizer
Visual: Eigenvectors (Blue) simply stretch or flip to become Red vectors. They do not rotate off their original line.
Understanding Diagonalization
Diagonalization is the process of finding a “better” coordinate system for a matrix. In standard coordinates, a matrix transformation might look like a complex mix of rotation and stretching. But if we switch to the Eigenbasis (a coordinate system made of eigenvectors), the transformation becomes a simple scaling along the axes.
The Formula: A = PDP⁻¹
- P (Change of Basis): A matrix containing the eigenvectors as columns. It translates standard coordinates into “eigen-coordinates”.
- D (Scaling): A diagonal matrix containing the eigenvalues. It performs the actual stretching.
- P⁻¹ (Return): Translates back to standard coordinates.
Why is it useful?
It makes calculating powers of matrices trivial. Instead of multiplying A by itself 100 times, you just power the diagonal numbers:
Ak = P Dk P⁻¹
What defines a Diagonalizable Matrix?
Not all matrices can be diagonalized. A matrix must have enough linearly independent eigenvectors to form a full basis (the matrix P must be invertible).
Common cases:
1. If an n x n matrix has n distinct eigenvalues, it is always diagonalizable.
2. Symmetric matrices (real entries) are always diagonalizable with orthogonal eigenvectors.
Frequently Asked Questions (FAQ)
Q: What if eigenvalues are complex?
A: The matrix is still diagonalizable, but the matrices P and D will contain complex numbers. Geometrically, this corresponds to a rotation in the complex plane.
Q: What is a “Defective” matrix?
A: A matrix that does not have enough eigenvectors to form P. This happens when repeated eigenvalues “overlap” too much (geometric multiplicity < algebraic multiplicity). These matrices cannot be diagonalized; we use Jordan Normal Form instead.
Q: Does the order of columns in P matter?
A: Yes, but only if you match them with D. If you put eigenvector v₁ in the first column of P, you must put eigenvalue λ₁ in the first column of D.