Linear algebra

Eigenvalue 2x2 Calculator

Enter the four entries of a 2x2 matrix to get its eigenvalues and eigenvectors, with every step of the working and a plot that shows how the matrix stretches the plane.

Matrix A
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EigenvalueEigenvectorUnit eigenvectorCheck A·v = λ·v
Step-by-step working

    Plot: grey is the unit circle, purple is its image under A, and the coloured lines are the real eigenvector directions, which A only stretches or flips.

    Runs entirely in your browser. Nothing is uploaded to any server.

    Why use this eigenvalue calculator

    Every case

    Distinct, repeated and complex eigenvalues, including defective matrices.

    Shows the working

    Characteristic equation, discriminant and eigenvector steps.

    See it

    A plot shows how the matrix stretches the unit circle.

    How to find eigenvalues of a 2x2 matrix

    1. Enter the matrixType a, b, c and d, or start from an example matrix.
    2. Read the eigenvaluesThey appear at the top with the trace, determinant and discriminant.
    3. Follow the stepsSee how each eigenvector is found and checked.

    The 2x2 eigenvalue formula

    For A = [[a, b], [c, d]], the eigenvalues solve det(A − λI) = 0, which for a 2x2 matrix is the quadratic λ² − (a + d)λ + (ad − bc) = 0. The trace a + d equals the sum of the eigenvalues and the determinant ad − bc equals their product.

    The discriminant Δ = (a + d)² − 4(ad − bc) decides the case. If Δ > 0 there are two real eigenvalues, if Δ = 0 there is one repeated eigenvalue, and if Δ < 0 the eigenvalues are complex conjugates, which happens for rotations. Each eigenvector is any non-zero vector perpendicular to a non-zero row of A − λI.

    Symmetric matrices

    Always have real eigenvalues and perpendicular eigenvectors.

    Rotation matrices

    Have complex eigenvalues because no direction stays fixed.

    Defective matrices

    A repeated eigenvalue with only one eigenvector, like a shear.

    Eigenvalue 2x2 FAQ

    What is a quick check for my answer?

    The two eigenvalues must add up to the trace a + d and multiply to the determinant ad − bc.

    Why are eigenvectors not unique?

    Any non-zero multiple of an eigenvector is also an eigenvector. The calculator shows a simple version and the unit-length version.

    What does a complex eigenvalue mean?

    The matrix rotates every direction, so no real vector keeps its direction. The real part controls growth or decay and the imaginary part the rotation speed.

    What is a defective matrix?

    One with a repeated eigenvalue but only one independent eigenvector, such as [[3, 1], [0, 3]]. It cannot be diagonalised.

    What is the equilibrium type used for?

    For the system of differential equations x' = Ax, the eigenvalues decide whether the origin is a node, saddle, spiral or centre and whether it is stable.

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