Matrix Diagonalization Calculator

Matrix diagonalization is the process of expressing a square matrix A as A = PDP⁻¹, where D is a diagonal matrix containing eigenvalues and P contains eigenvectors. This is crucial for simplifying matrix computations, solving differential equations, and understanding linear transformations. For numerical calculations, you can also use our percentage calculator for related mathematical operations.

A matrix is diagonalizable if it has n linearly independent eigenvectors (for an n×n matrix). All symmetric matrices are diagonalizable. If a matrix has distinct eigenvalues, it's guaranteed to be diagonalizable. However, matrices with repeated eigenvalues or complex eigenvalues may not be diagonalizable over real numbers.

An eigenvalue (λ) is a scalar that satisfies the equation Av = λv, where v is a non-zero eigenvector. Eigenvectors are special vectors that only get scaled (not rotated) when a linear transformation is applied. They represent the fundamental directions of the transformation, while eigenvalues represent the scaling factors.

For a 2×2 matrix, solve the characteristic equation det(A - λI) = 0. This gives you λ² - (trace)λ + det = 0, which can be solved using the quadratic formula: λ = (trace ± √(trace² - 4·det)) / 2. For exponential calculations in matrix operations, our logarithm calculator can be helpful.

The characteristic polynomial is obtained from det(A - λI) = 0. For an n×n matrix, it's a polynomial of degree n whose roots are the eigenvalues. For a 2×2 matrix, it's a quadratic equation, and for a 3×3 matrix, it's a cubic equation. Solving this polynomial gives you all eigenvalues.

No, not all matrices are diagonalizable. A matrix must have n linearly independent eigenvectors to be diagonalizable. Matrices with complex eigenvalues cannot be diagonalized over real numbers. Defective matrices (those with repeated eigenvalues and insufficient eigenvectors) are also not diagonalizable.

In the diagonalization formula A = PDP⁻¹, P is the matrix whose columns are eigenvectors, D is a diagonal matrix with eigenvalues on the diagonal, and P⁻¹ is the inverse of P. This decomposition allows complex matrix operations to be simplified by working with the diagonal matrix D instead.

Diagonalization makes computing matrix powers much easier. If A = PDP⁻¹, then Aⁿ = PDⁿP⁻¹. Since D is diagonal, Dⁿ is simply obtained by raising each diagonal element to the nth power, which is computationally efficient. You can use our curl calculator for other vector calculus operations.

When eigenvalues are repeated, the matrix may or may not be diagonalizable. It depends on whether there are enough linearly independent eigenvectors. If an eigenvalue λ has multiplicity k (appears k times), you need k linearly independent eigenvectors for that eigenvalue to diagonalize the matrix. Otherwise, the matrix is defective and not diagonalizable.

Matrix diagonalization has numerous applications: quantum mechanics (observable operators), vibration analysis in engineering, population dynamics modeling, Google's PageRank algorithm, Principal Component Analysis (PCA) in machine learning, solving systems of differential equations, image compression, and stability analysis in control theory. These applications leverage the computational efficiency of working with diagonal matrices.