Matrix Diagonalization Calculator

Calculate eigenvalues, eigenvectors, and diagonalization of matrices. Step-by-step linear algebra tool for students and professionals.

Diagonalization Formula: A square matrix A is diagonalizable if there exists an invertible matrix P and a diagonal matrix D such that:

A = PDP⁻¹ or D = P⁻¹AP

Where D contains the eigenvalues of A, and P contains the corresponding eigenvectors as columns.

2×2
3×3
4×4
5×5
6×6
Advanced Options
Diagonalizable 2×2
Symmetric 2×2
Complex Eigenvalues
Diagonalizable 3×3
Symmetric 3×3
Identity Matrix
Non-diagonalizable
Rotation Matrix

Understanding Matrix Diagonalization

Diagonalization is the process of finding a diagonal matrix D that is similar to a given square matrix A. A matrix is diagonalizable if it can be expressed in the form A = PDP⁻¹, where D is a diagonal matrix and P is an invertible matrix.

Mathematical Definition:

An n×n matrix A is diagonalizable if there exists an invertible matrix P and a diagonal matrix D such that:

A = PDP⁻¹

The diagonal entries of D are the eigenvalues of A, and the columns of P are the corresponding eigenvectors.

Conditions for Diagonalizability

1

Eigenvalue Condition: An n×n matrix is diagonalizable if and only if it has n linearly independent eigenvectors.

2

Algebraic vs. Geometric Multiplicity: For each eigenvalue, its geometric multiplicity (dimension of eigenspace) must equal its algebraic multiplicity (multiplicity as a root of the characteristic polynomial).

3

Special Cases: A symmetric matrix is always diagonalizable. A matrix with n distinct eigenvalues is always diagonalizable.

Steps for Diagonalization

Step Description Example
1 Find the eigenvalues λ by solving det(A - λI) = 0 For A = [[2,1],[1,2]], eigenvalues are λ=1,3
2 For each eigenvalue, find eigenvectors by solving (A - λI)v = 0 For λ=1: v₁=[1,-1]; for λ=3: v₂=[1,1]
3 Construct matrix P with eigenvectors as columns P = [[1,1],[-1,1]]
4 Construct diagonal matrix D with eigenvalues on diagonal D = [[1,0],[0,3]]
5 Verify that A = PDP⁻¹ or D = P⁻¹AP A = PDP⁻¹

Applications of Diagonalization

  • Matrix Powers: Aᵏ = PDᵏP⁻¹ makes computing matrix powers efficient
  • Differential Equations: Solving systems of linear differential equations
  • Markov Chains: Analyzing steady-state behavior
  • Quantum Mechanics: Finding energy states and observables
  • Principal Component Analysis (PCA): Dimensionality reduction in data science
  • Vibration Analysis: Finding normal modes in mechanical systems

Calculator Features:

  • Calculates eigenvalues and eigenvectors of matrices up to 6×6
  • Supports complex eigenvalues and eigenvectors
  • Determines if a matrix is diagonalizable
  • Computes the diagonalization A = PDP⁻¹ when possible
  • Shows step-by-step calculations including characteristic polynomial
  • Matrix power calculator for diagonalizable matrices
  • Export results as text, image, or Excel
  • Calculation history and advanced options

Frequently Asked Questions

A diagonalizable matrix has a complete set of linearly independent eigenvectors, allowing it to be expressed as A = PDP⁻¹ where D is diagonal. A non-diagonalizable matrix (defective matrix) does not have enough linearly independent eigenvectors to form the matrix P. The most common example is a matrix with repeated eigenvalues where the geometric multiplicity is less than the algebraic multiplicity.

Yes, all real symmetric matrices are diagonalizable. In fact, they are orthogonally diagonalizable, meaning there exists an orthogonal matrix P (with P⁻¹ = Pᵀ) such that A = PDPᵀ. This is a consequence of the Spectral Theorem for symmetric matrices.

This calculator fully supports complex eigenvalues and eigenvectors. Complex eigenvalues appear in conjugate pairs for real matrices. The calculator will display complex numbers in the form a + bi, where a is the real part and b is the imaginary part.

Diagonalization simplifies many matrix computations. For example, computing powers of matrices becomes much easier: Aᵏ = PDᵏP⁻¹, where Dᵏ is just the diagonal matrix with each diagonal entry raised to the kth power. Diagonalization also helps in solving systems of linear differential equations, analyzing dynamical systems, and understanding the geometric transformation represented by the matrix.

Yes, a matrix with repeated eigenvalues can still be diagonalizable if the geometric multiplicity of each eigenvalue equals its algebraic multiplicity. For example, the identity matrix has all eigenvalues equal to 1 (repeated n times for an n×n matrix), but it is diagonalizable because it already is a diagonal matrix. However, if the geometric multiplicity is less than the algebraic multiplicity for any eigenvalue, the matrix is not diagonalizable.