How to use
Enter a, b and d for A = [a b; b d]. The off-diagonal value b is shared, so the matrix is symmetric by construction. Calculate to confirm both eigenpairs. The live unit-circle diagram follows valid edits; the arrows show normalized directions, not eigenvalue magnitudes.
Read the result
Each eigenpair satisfies Av = λv. Positive λ stretches along the same direction; negative λ reverses it; zero maps the vector to zero. Unit eigenvectors have length 1 and are perpendicular. λ₁ ≥ λ₂; displayed vectors use extra digits so their directions remain useful. The reported relative residual uses unrounded values, divided by the largest input magnitude.
Example
For 3 1; 1 3, λ₁ = 4 and λ₂ = 2. Unit vectors are approximately (0.707107, 0.707107) and (-0.707107, 0.707107). Formula: λ = (a+d)/2 ± √(((a-d)/2)²+b²). With A = 3I, both eigenvalues are 3 and the coordinate axes are one valid basis.
Limits and errors
Only real symmetric 2×2 matrices. Use decimal points or e notation. Each nonzero input must have magnitude 10⁻⁹–10⁹. Double-precision calculations can lose relative accuracy for very small eigenvalues or nearly repeated roots. Values are approximate; a small residual is a consistency check, not a guarantee of accurate data.
Frequently asked questions
Why does another answer use the opposite vector?
v and −v describe the same eigendirection and are both correct. For A = aI, including the zero matrix, every direction is an eigenvector direction; the displayed perpendicular axes are an arbitrary valid choice. Nearly repeated eigenvalues can make directions sensitive to tiny input changes.
Can I edit or copy the result?
Editing invalidates the confirmed result and disables Copy. Calculate again before copying. Clear leaves the fields empty. The pale example runs immediately; the first focus in a value field clears all example values once.
Sources: TU Delft · Linear algebra