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2x2 Matrix Eigenvalues Calculator

Find the eigenvalues of a 2x2 matrix [[a,b],[c,d]] by solving its characteristic equation λ² − trace·λ + determinant = 0.

Trace (a + d)
5
Determinant (ad − bc)
6
Nature
Two distinct real eigenvalues
Eigenvalue λ1
3
Eigenvalue λ2
2

y = ax² + bx + c

The curve crosses the x-axis at each real root.

How it works

For a 2x2 matrix [[a,b],[c,d]], the eigenvalues are the roots of its characteristic polynomial, found by solving λ² − trace·λ + determinant = 0 where trace = a + d and determinant = ad − bc. That's a plain quadratic in λ, so the quadratic formula gives both eigenvalues directly: λ = (trace ± √(trace² − 4·determinant)) / 2.

The quantity under the square root, trace² − 4·determinant, decides what kind of eigenvalues the matrix has — exactly as a discriminant does for any quadratic. Positive gives two distinct real eigenvalues, zero gives one repeated real eigenvalue, and negative gives a complex-conjugate pair (a matrix like a rotation, which has no real direction it simply stretches, is the standard example of this last case).

The chart below plots this same characteristic polynomial as a parabola in λ — where it crosses zero is exactly where the eigenvalues are, so a parabola that dips below the axis has two real eigenvalues, one that just touches the axis has a repeated eigenvalue, and one that stays entirely above the axis has a complex-conjugate pair instead.

FAQ

What does an eigenvalue actually mean?

An eigenvalue λ is a scalar for which some nonzero vector v (the eigenvector) satisfies Mv = λv — applying the matrix to that particular direction just stretches or shrinks it by λ, without rotating it. A 2x2 matrix has exactly two eigenvalues (counted with repetition), which together summarize how the matrix scales space along its two special directions.

Why do some matrices have complex eigenvalues?

A real matrix can still have a negative discriminant, just like a real quadratic can, and then its eigenvalues are a complex-conjugate pair rather than real numbers — geometrically this happens for matrices that rotate the plane rather than purely stretching it, since no real direction comes back out as a plain scalar multiple of itself under a rotation.

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