Diagonalizing a symmetric 3×3 matrix

A worked diagonalization of the symmetric 3×3 matrix with rows (2, 1, 0), (1, 2, 0), (0, 0, 3). See the characteristic polynomial, the eigenvalues and eigenvectors, and whether A = PDP⁻¹ exists.

Every real symmetric matrix is diagonalizable, and its eigenvectors can be chosen orthogonal. The calculator below is loaded with the 3x3 matrix with rows (2, 1, 0), (1, 2, 0), (0, 0, 3).

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. defective (non-diagonalizable) matrix A repeated eigenvalue with only one eigenvector means no basis of eigenvectors exists — this matrix is defective. 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 symmetric 3×3 matrix be diagonalized? ▾

Every real symmetric matrix is diagonalizable, and its eigenvectors can be chosen orthogonal.

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 symmetric 3×3 matrix with rows (2, 1, 0), (1, 2, 0), (0, 0, 3). See the characteristic polynomial, the eigenvalues and eigenvectors, and whether A = PDP⁻¹ exists.

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