Diagonalizing a defective (non-diagonalizable) matrix

A worked diagonalization of the defective (non-diagonalizable) matrix with rows (2, 1), (0, 2). See the characteristic polynomial, the eigenvalues and eigenvectors, and whether A = PDP⁻¹ exists.

A repeated eigenvalue with only one eigenvector means no basis of eigenvectors exists — this matrix is defective. The calculator below is loaded with the 2x2 matrix with rows (2, 1), (0, 2).

Matrix A — fractions like 3/4 are fine

More diagonalization examples

2×2 matrix with distinct eigenvalues Distinct eigenvalues always give independent eigenvectors, so this matrix is diagonalizable. symmetric 3×3 matrix Every real symmetric matrix is diagonalizable, and its eigenvectors can be chosen orthogonal. already diagonal matrix A diagonal matrix is trivially diagonalizable: P is the identity and D is the matrix itself. matrix with complex eigenvalues A rotation has no real eigenvectors, so it cannot be diagonalized over the reals.

Frequently asked questions

Can this defective (non-diagonalizable) matrix be diagonalized? ▾

A repeated eigenvalue with only one eigenvector means no basis of eigenvectors exists — this matrix is defective.

How do I find the eigenvalues myself? ▾

Form A − λI, take its determinant to get the characteristic polynomial, and solve for λ. The calculator shows the polynomial it found so you can compare with your own working.

What do I do with the eigenvectors once I have them? ▾

Place them as the columns of P, keeping the same order as the eigenvalues on D's diagonal. Then A = PDP⁻¹, and the calculator verifies the product for you.

About this Calculator

A worked diagonalization of the defective (non-diagonalizable) matrix with rows (2, 1), (0, 2). See the characteristic polynomial, the eigenvalues and eigenvectors, and whether A = PDP⁻¹ exists.

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