What this calculator does
An eigenvector calculator finds the special directions of a matrix that do not get rotated when the matrix acts on them, only stretched or shrunk. For a 2×2 matrix, that means solving one quadratic equation for the two eigenvalues, then a small linear system for each matching eigenvector.
This tool works with a 2×2 matrix only. Larger matrices need iterative numerical methods to find eigenvalues reliably, which is a different kind of computation to the exact quadratic-formula solution used here, so extending this approach past 2×2 would trade accuracy for false confidence.
The formula
Enter the four entries a, b, c and d of the matrix [[a, b], [c, d]]. The eigenvalues solve the characteristic equation λ² − (a+d)λ + (ad−bc) = 0, where a+d is the trace and ad−bc is the determinant. The quadratic formula gives λ = (trace ± √discriminant) / 2, where discriminant = trace² − 4×determinant. Once an eigenvalue is known, its eigenvector solves (A − λI)v = 0: if b is not zero, the eigenvector is (b, λ−a); otherwise if c is not zero, it is (λ−d, c).
| Term | Meaning |
|---|---|
| Eigenvalue (λ) | A scalar such that Av = λv for some nonzero vector v: the amount that vector is stretched or shrunk, with no change of direction. |
| Eigenvector | A nonzero vector v whose direction is unchanged when multiplied by the matrix; only its length (and possibly sign) changes. |
| Trace | The sum of the diagonal entries, a + d. |
| Determinant | ad − bc, which is zero exactly when the matrix has no inverse. |
| Discriminant | trace² − 4×determinant. Positive gives two distinct real eigenvalues, zero gives one repeated real eigenvalue, negative gives a complex-conjugate pair. |
The inputs explained
| Field | What to enter |
|---|---|
| Matrix entry a (row 1, col 1) | Top-left entry of the matrix. |
| Matrix entry b (row 1, col 2) | Top-right entry of the matrix. |
| Matrix entry c (row 2, col 1) | Bottom-left entry of the matrix. |
| Matrix entry d (row 2, col 2) | Bottom-right entry of the matrix. |
When to use it
Checking a hand-worked linear algebra problem
A characteristic-equation and eigenvector calculation by hand has several places to slip, particularly a sign error solving (A − λI)v = 0. Running the same matrix through here checks the answer independently.
Understanding what a transformation does to a shape
The eigenvectors of a 2×2 matrix are the only two directions (when real and distinct) that a linear transformation stretches without rotating, which is the geometric picture behind the algebra.
Spotting a repeated or complex case before it causes confusion
A repeated eigenvalue can mean the matrix is defective, with only one independent eigenvector direction rather than two, and a negative discriminant means there are no real eigenvectors at all. Both are easy to miss when working by hand and are flagged automatically here.
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 the eigenvalues change as the top-left entry a varies
The default matrix [[a, 1], [2, 3]], with a varied across a range of values.
| Entry a | Eigenvalue λ1 | Eigenvalue λ2 | Determinant (ad−bc) |
|---|---|---|---|
| 1 | 3.73205081 | 0.26794919 | 1 |
| 2 | 4 | 1 | 4 |
| 3 | 4.41421356 | 1.58578644 | 7 |
| 4 | 5 | 2 | 10 |
| 5 | 5.73205081 | 2.26794919 | 13 |
| 6 | 6.56155281 | 2.43844719 | 16 |
How the eigenvalues change as the off-diagonal entry b varies
The default matrix [[4, b], [2, 3]], with b varied across a range of values, including the point where the discriminant crosses zero.
| Entry b | Eigenvalue λ1 | Eigenvalue λ2 | Discriminant |
|---|---|---|---|
| -2 | 3.5 + 1.93649167i | 3.5 − 1.93649167i | -15 |
| -1 | 3.5 + 1.32287566i | 3.5 − 1.32287566i | -7 |
| 0 | 4 | 3 | 1 |
| 1 | 5 | 2 | 9 |
| 2 | 5.56155281 | 1.43844719 | 17 |
| 4 | 6.37228132 | 0.62771868 | 33 |
Questions
Why does this calculator only handle 2×2 matrices?
A 2×2 matrix has a quadratic characteristic equation, solvable exactly with the quadratic formula. Matrices of 3×3 or larger generally need iterative numerical methods to find eigenvalues, which is a fundamentally different and less exact kind of calculation than the one used here.
What does it mean if the eigenvalues come out complex?
A negative discriminant means the matrix has no real eigenvectors at all: geometrically, it rotates every vector rather than just stretching some special set of directions. The matrix still has two eigenvalues, but they form a complex-conjugate pair rather than real numbers.
What does a repeated eigenvalue mean for the eigenvectors?
If the matrix is a multiple of the identity (b = c = 0 and a = d), every nonzero vector is an eigenvector. Otherwise a repeated eigenvalue usually means the matrix is defective: it only has one independent eigenvector direction instead of two, which matters for whether the matrix can be diagonalised.
Why are eigenvectors given as a ratio rather than a single fixed vector?
Any nonzero multiple of an eigenvector is also an eigenvector for the same eigenvalue, since scaling a vector does not change its direction. The values shown here are one valid solution to (A − λI)v = 0; multiplying or dividing both entries by the same number gives an equally correct eigenvector.
For the building blocks behind this calculation, the matrix operations calculator and the determinant calculator cover the underlying matrix arithmetic on their own.