Matrix Diagonalization Calculator

How it is calculated

For the matrix [[a, b], [c, d]] the tool computes the trace tr = a + d and determinant det = ad − bc. The eigenvalues are the roots of λ² − tr·λ + det = 0, that is λ = (tr ± √(tr² − 4·det)) ÷ 2. The larger root is shown first. If the discriminant is negative the eigenvalues are complex and the tool returns no result. A repeated eigenvalue is accepted only for a scalar matrix (b = c = 0 and a = d); any other matrix with a repeated eigenvalue cannot be diagonalized and shows nothing. Each eigenvector is taken as [b, λ − a] (or [λ − d, c] when b is zero), left unnormalised and printed to 4 decimals; a diagonal matrix gives [1, 0] and [0, 1]. The columns of P are the eigenvectors and D has the eigenvalues on its diagonal, so A = P·D·P⁻¹. You assemble P and D yourself. Scaled copies of an eigenvector are equally valid.

Common mistakes

• Feeding in a rotation-like matrix such as [[0, −1], [1, 0]]. Its eigenvalues are complex, so the tool returns nothing; it works only with real eigenvalues.
• Entering the entries in column order instead of row order. The fields are a, b (top row) and c, d (bottom row); swapping b and c transposes the matrix and changes the eigenvectors.
• Expecting an answer for a shear such as [[1, 1], [0, 1]]. It has a repeated eigenvalue but only one independent eigenvector, so it is not diagonalizable and the tool shows no result. Only the identity-like scalar matrix [[3, 0], [0, 3]] passes with a repeated value.
• Expecting one fixed eigenvector. Any non-zero multiple is correct, so [1, 1] and [2, 2] describe the same direction, and the tool does not normalise to unit length.
• Ignoring the order of the eigenvalues. The larger one is always listed first, so on [[2, 0], [0, 5]] the first eigenvector is [0, 1] and the second is [1, 0]; build P and D in the same order.