Matrix Diagonalization Calculator
Diagonalize a square matrix by computing eigenvalues, eigenvectors, and the decomposition A = PDP⁻¹. Supports 2×2 to 5×5 matrices with step-by-step solutions, characteristic polynomial, multiplicity analysis, and interactive visualization.
Enter a square matrix below to get its eigenvalues, the matrices P and D with A = PDP⁻¹, and the steps.
About Matrix Diagonalization Calculator
The Matrix Diagonalization Calculator decomposes any square matrix into the form A = PDP⁻¹, where D is a diagonal matrix of eigenvalues and P is the matrix of eigenvectors. Enter a 2×2 to 5×5 matrix and get the complete factorization with step-by-step solutions, characteristic polynomial, algebraic and geometric multiplicity analysis, and interactive animation of the decomposition.
What Is Matrix Diagonalization?
Matrix diagonalization is the process of finding matrices P and D such that:
$$A = PDP^{-1}$$
where D is a diagonal matrix whose entries are the eigenvalues of A, and P is an invertible matrix whose columns are the corresponding eigenvectors. Equivalently, \(D = P^{-1}AP\), which means D is similar to A.
How to Diagonalize a Matrix
Step 1. Select the matrix size (2×2 to 5×5) and enter values into the grid. You can also click a quick example to load a preset matrix for testing.
Step 2. Click Diagonalize Matrix. The calculator computes the characteristic polynomial det(A − λI) and finds its roots (eigenvalues).
Step 3. For each eigenvalue, the tool solves (A − λI)x = 0 to find eigenvectors, and checks algebraic vs. geometric multiplicity to determine if the matrix is diagonalizable.
Step 4. If diagonalizable, the calculator constructs P (eigenvectors as columns), D (eigenvalues on diagonal), and P⁻¹, then verifies PDP⁻¹ = A.
Step 5. Explore the animated decomposition to visualize how A factors into P × D × P⁻¹, and step through the full solution using the navigation controls.
When Is a Matrix Diagonalizable?
| Condition | Diagonalizable? | Example |
|---|---|---|
| n distinct real eigenvalues | Always yes | \(\begin{pmatrix}2&1\\0&3\end{pmatrix}\) → λ = 2, 3 |
| Symmetric matrix (A = Aᵀ) | Always yes (real λ) | Spectral theorem guarantees orthogonal diagonalization |
| Repeated λ with AM = GM | Yes | \(\begin{pmatrix}5&0\\0&5\end{pmatrix}\) → λ = 5 (AM=2, GM=2) |
| Repeated λ with AM > GM | No | \(\begin{pmatrix}1&1\\0&1\end{pmatrix}\) → λ = 1 (AM=2, GM=1) |
| Complex eigenvalues | Over ℂ: check AM = GM | \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) → λ = ±i |
Algebraic vs. Geometric Multiplicity
For each eigenvalue λ:
• Algebraic multiplicity (AM): the number of times λ appears as a root of the characteristic polynomial det(A − λI) = 0.
• Geometric multiplicity (GM): the dimension of the eigenspace ker(A − λI), i.e., the number of linearly independent eigenvectors.
A matrix is diagonalizable if and only if GM = AM for every eigenvalue. The condition 1 ≤ GM ≤ AM always holds.
Why Diagonalization Matters
Diagonalization vs. Other Decompositions
| Decomposition | Form | Requirement |
|---|---|---|
| Eigendecomposition (this tool) | A = PDP⁻¹ | n independent eigenvectors |
| Spectral (symmetric) | A = QΛQᵀ | A = Aᵀ (Q orthogonal) |
| Jordan Normal Form | A = PJP⁻¹ | Any square matrix |
| SVD | A = UΣVᵀ | Any matrix (even non-square) |
| LU Decomposition | A = LU | Square, with conditions |
Frequently Asked Questions
What does it mean to diagonalize a matrix?
Diagonalizing a matrix A means finding an invertible matrix P and a diagonal matrix D such that A = PDP⁻¹. The diagonal entries of D are the eigenvalues, and the columns of P are the corresponding eigenvectors.
When is a matrix diagonalizable?
A matrix is diagonalizable if and only if, for every eigenvalue, the geometric multiplicity equals the algebraic multiplicity. Equivalently, there must be n linearly independent eigenvectors for an n×n matrix. All symmetric real matrices and all matrices with n distinct eigenvalues are diagonalizable.
What is the difference between algebraic and geometric multiplicity?
The algebraic multiplicity is how many times an eigenvalue appears as a root of the characteristic polynomial. The geometric multiplicity is the dimension of the eigenspace, i.e., the number of linearly independent eigenvectors for that eigenvalue. A matrix is diagonalizable precisely when these two quantities are equal for every eigenvalue.
Can a matrix with complex eigenvalues be diagonalized?
Yes, a matrix with complex eigenvalues can still be diagonalized over the complex numbers, as long as the geometric multiplicity equals the algebraic multiplicity for each eigenvalue. The resulting P and D matrices will contain complex entries.
What are the applications of matrix diagonalization?
Matrix diagonalization is used to compute matrix powers efficiently (A^k = PD^kP⁻¹), solve systems of differential equations, analyze Markov chains and steady-state behavior, perform principal component analysis in statistics, and understand linear transformations in physics and engineering.
Reference this content, page, or tool as:
"Matrix Diagonalization Calculator" at https://MiniWebtool.com/matrix-diagonalization-calculator/ from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-04-12
Linear Algebra:
- Angle Between Vectors Calculator New
- Characteristic Polynomial Calculator New
- Cholesky Decomposition Calculator New
- Column Space Calculator New
- Cramer's Rule Calculator New
- Cross Product Calculator New
- Determinant Calculator
- Dot Product Calculator New
- Eigenvalue and Eigenvector Calculator
- Gram-Schmidt Calculator
- Jacobian Matrix Calculator New
- Jordan Normal Form Calculator New
- Matrix Calculator
- Matrix Diagonalization Calculator New
- Matrix Exponential Calculator New
- Matrix Inverse Calculator New
- Matrix LU Decomposition Calculator
- Matrix Multiplication Calculator New
- Matrix Power Calculator New
- Matrix Rank Calculator
- Matrix Trace Calculator
- Null Space Calculator New
- Partial Fraction Decomposition Calculator
- QR Decomposition Calculator New
- RREF Calculator (Row Echelon Form) New
- Singular Value Decomposition (SVD) Calculator
- Tensor Product Calculator New
- Unit Vector Calculator New
- Vector Calculator
- Vector Magnitude Calculator New
- Vector Projection Calculator