Diagonalize a 2×2 Matrix
Diagonalize any 2×2 matrix step by step. Get the degree-2 characteristic polynomial, all 2 eigenvalues, the eigenvectors, and the matrices P and D with A = PDP⁻¹.
Enter a 2×2 matrix to get its characteristic polynomial, its 2 eigenvalues, a basis of eigenvectors for each, and the factorisation A = PDP⁻¹ with a verification step.
Two quick checks on your eigenvalues
For any 2×2 matrix, the 2 eigenvalues must add up to the trace (the sum of the diagonal entries) and multiply to the determinant. Both are worth checking before you go on to find eigenvectors — they catch most arithmetic slips in the characteristic polynomial.
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Frequently asked questions
How do you diagonalize a 2×2 matrix? ▾
Solve det(A − λI) = 0, which is a degree-2 polynomial, to get the eigenvalues. For each one, find the null space of A − λI to get its eigenvectors. If you end up with 2 independent eigenvectors, they form the columns of P and the eigenvalues form D.
Is every 2×2 matrix diagonalizable? ▾
No. A 2×2 matrix is diagonalizable only if it has 2 linearly independent eigenvectors. 2 distinct real eigenvalues guarantee it, but a repeated eigenvalue with a single eigenvector (a defective matrix) or complex eigenvalues will block it over the reals.
What degree is the characteristic polynomial of a 2×2 matrix? ▾
Degree 2. Its roots are the eigenvalues, its linear coefficient relates to the trace, and its constant term relates to the determinant.
How do you check a 2×2 diagonalization? ▾
Multiply P by D by P⁻¹. If the product is your original matrix, the diagonalization is correct — the calculator runs this check and displays the result.
About this Calculator
Diagonalize any 2×2 matrix step by step. Get the degree-2 characteristic polynomial, all 2 eigenvalues, the eigenvectors, and the matrices P and D with A = PDP⁻¹.