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Eigendecomposition rewrites a matrix A as A = PDP⁻¹, where D is a diagonal matrix of eigenvalues and P's columns are the corresponding eigenvectors. For a symmetric matrix, P is guaranteed orthogonal, which means P⁻¹ is simply Pᵀ — no separate matrix inversion is ever needed.
This calculator computes the full A = PDP⁻¹ decomposition for a symmetric matrix up to 6×6, showing P, D, and P⁻¹ (= Pᵀ) explicitly.
How it works
Enter a symmetric matrix. The tool uses the same Jacobi rotation method as the Symmetric Matrix Eigenvalue Calculator to find the eigenvalues (forming D) and eigenvectors (forming the columns of P), then reports P's transpose as P⁻¹.
- Click Calculate to see your results.
Examples
A 2×2 symmetric example
For A = [[2,1],[1,2]], eigendecomposition gives D = diag(3,1) and P with columns (1,1)/√2 and (1,−1)/√2 — and multiplying P·D·Pᵀ back together reconstructs A exactly.
Who should use it
- Diagonalizing a symmetric matrix for use in further calculations (like computing A raised to a power, which becomes trivial once diagonalized).
- Understanding the geometric structure of a symmetric transformation (its principal axes and scaling factors).
Industry applications
- Physics and engineering (principal axes, moment of inertia)
- Statistics and data science (principal component analysis)
Advantages
- Shows the complete decomposition (P, D, and P⁻¹) rather than just eigenvalues.
- P⁻¹ = Pᵀ is always exactly true for a symmetric input, so no matrix inversion error can creep in.
Limitations
- Only applicable to symmetric matrices.
Common mistakes to avoid
- Feeding in a non-symmetric matrix, where P is not guaranteed to be orthogonal (or for some matrices, to exist at all in real form).
Best practices
- Verify your result by multiplying P·D·P⁻¹ back together and confirming it reconstructs your original matrix A.
Tips
- Once you have A = PDP⁻¹, computing A raised to any power n is easy: Aⁿ = PDⁿP⁻¹, and Dⁿ is just each diagonal entry raised to the power n.