Diagonalize Matrix Calculator

Diagonalizing a matrix A means finding matrices P and D such that A = PDP⁻¹, where D is a diagonal matrix. P is formed from the eigenvectors of A as its columns, and D has the corresponding eigenvalues along its main diagonal. This form makes repeated matrix powers and other operations much simpler to compute.

A matrix is diagonalizable if and only if it has enough linearly independent eigenvectors to form a complete basis — specifically, the sum of the geometric multiplicities of all eigenvalues must equal n (the matrix size). If a repeated eigenvalue has fewer independent eigenvectors than its algebraic multiplicity, the matrix is called defective and cannot be diagonalized.

Eigenvalues are the values λ satisfying det(A − λI) = 0, known as the characteristic equation. Each eigenvalue has at least one associated eigenvector. In the diagonalized form A = PDP⁻¹, the eigenvalues appear as the diagonal entries of D, making them the 'simplest representation' of how A scales space.

Matrix P (the modal matrix) is constructed by placing the eigenvectors of A as its columns in order. Matrix D (the diagonal matrix) contains the corresponding eigenvalues on its main diagonal, in the same order as the eigenvectors in P. Together they satisfy A = PDP⁻¹.

Yes — every real symmetric matrix is diagonalizable (this follows from the Spectral Theorem). Moreover, symmetric matrices can always be orthogonally diagonalized, meaning the matrix P can be chosen to be orthogonal (P⁻¹ = Pᵀ). This is a very useful property in physics and engineering applications.

Select the size of your square matrix (2×2, 3×3, or 4×4), then enter each entry of the matrix in the corresponding fields. The calculator automatically finds the eigenvalues, constructs the eigenvector matrix P and the diagonal matrix D, and tells you whether the matrix is diagonalizable.

Diagonalization is widely used in differential equations, quantum mechanics, computer graphics, principal component analysis (PCA), Markov chains, and structural engineering. Because diagonal matrices are computationally simple, diagonalizing A makes it straightforward to compute Aⁿ, matrix exponentials, and solve linear systems.

This calculator focuses on real eigenvalues arising from real matrices. Matrices with complex eigenvalues (such as rotation matrices) are not diagonalizable over the real numbers but can be diagonalized over the complex numbers. For such cases the calculator will report the matrix as not diagonalizable in the real domain.