2 × 2 Matrix Eigenvalues

Find real or conjugate-complex eigenvalues of a two-by-two matrix.

Your numbers

First real part

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Use consistent units. The result follows the assumptions you enter.

Formula

For the matrix [[a,b],[c,d]], use λ = (a+d±√((a−d)²+4bc))/2. A negative discriminant gives conjugate complex eigenvalues; the result lists real and imaginary parts separately.

Find real or conjugate-complex eigenvalues of a two-by-two matrix.

Worked example

Row 1, column 1: 2
Row 1, column 2: 1
Row 2, column 1: 1
Row 2, column 2: 2

First real part: 3
First imaginary part: 0
Second real part: 1
Second imaginary part: 0

Scope and limitations

Use consistent units. The result follows the assumptions you enter.

Inputs are checked for missing values and allowed ranges. A result outside the finite numeric range is not displayed. Rounding affects display, while the calculation uses the model’s numeric precision.

Check your units and assumptions before relying on a result. Changing a label does not convert a currency or physical unit.

Formula version: 9c6fa44aa0a7 · Example checked: 2026-10-30

Methodology