What this calculator does
An eigenvalue is a scaling factor along a direction the matrix leaves unchanged. Finding them means solving a quadratic built from just two numbers: the trace and the determinant.
The discriminant decides whether they are real. The matrix with rows 2, 1 and 1, 2 gives eigenvalues of 3.000 and 1.000, while a rotation matrix gives a complex pair, which makes sense: a rotation leaves no real direction unchanged.
The formula
The characteristic polynomial is lambda squared minus the trace times lambda plus the determinant. Solving that quadratic gives the two eigenvalues.
| Term | Meaning |
|---|---|
| Eigenvalue | A factor by which the matrix scales one of its eigenvectors. |
| Characteristic polynomial | The quadratic whose roots are the eigenvalues. |
| Complex pair | What results when the discriminant is negative, indicating rotation. |
The inputs explained
| Field | What to enter |
|---|---|
| Row 1, Col 1 (a) | Row 1, column 1. |
| Row 1, Col 2 (b) | Row 1, column 2. |
| Row 2, Col 1 (c) | Row 2, column 1. |
| Row 2, Col 2 (d) | Row 2, column 2. |
When to use it
Analysing a transformation
Eigenvalues describe how the matrix stretches along its special directions.
Checking stability
In dynamical systems the eigenvalues determine whether a fixed point is stable.
Understanding rotation
A complex pair indicates the transformation rotates rather than stretches.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the off-diagonal entry change things?
The same diagonal with three off-diagonal values.
Questions
What does an eigenvalue mean geometrically?
It is the factor by which the matrix stretches vectors along one particular direction. Most directions get rotated as well as scaled; eigenvectors are the exceptional directions that only get scaled.
What do complex eigenvalues indicate?
Rotation. If no real direction survives the transformation unchanged, there can be no real eigenvalue, and a rotation matrix is the clearest example. The complex pair encodes the rotation angle and scaling.
How do trace and determinant relate to the eigenvalues?
The eigenvalues sum to the trace and multiply to the determinant. That is a useful check: if your two eigenvalues do not reproduce both, something has gone wrong.
What does a zero eigenvalue mean?
That the matrix is singular. A zero eigenvalue means some direction gets collapsed entirely, which is exactly what a zero determinant describes, since the eigenvalues multiply to the determinant.
For the determinant on its own, see the determinant calculator. For the trace, see the matrix trace and transpose calculator.