2x2 Matrix Eigenvalue Calculator
Eigenvalues are the scalars by which a linear transformation stretches its special directions, the eigenvectors. For a 2x2 matrix they are the two roots of the characteristic equation, a simple quadratic built from the matrix trace and determinant. This calculator takes the four entries a, b, c, d, forms the trace and determinant, evaluates the discriminant, and reports the eigenvalues. When the discriminant is negative it correctly returns the complex conjugate pair with separate real and imaginary parts rather than a meaningless real value.
Eigenvalue formula
trace T = a + d
determinant D = a*d - b*c
discriminant = T^2 - 4D
lambda = ( T +/- sqrt(discriminant) ) / 2
complex if discriminant < 0
The eigenvalues solve lambda squared minus T lambda plus D equals zero. When the discriminant is negative the roots are complex conjugates with real part T over two and imaginary part the square root of the negated discriminant over two.
Notes on eigenvalues
- The sum of the eigenvalues equals the trace a + d.
- The product of the eigenvalues equals the determinant a*d - b*c.
- A negative discriminant gives a complex conjugate pair, shown with real and imaginary parts.
- A zero discriminant gives one repeated eigenvalue.
- A symmetric real matrix (b = c) always has real eigenvalues.
Eigenvalues: frequently asked questions
What is an eigenvalue?
An eigenvalue is a scalar lambda such that the matrix A times some nonzero vector v equals lambda times v. The vector v is the corresponding eigenvector. Eigenvalues describe the directions a linear transformation merely stretches without rotating.
How are 2x2 eigenvalues computed?
For a matrix [[a, b], [c, d]], the eigenvalues solve the characteristic equation lambda squared minus (trace) times lambda plus (determinant) equals zero, where trace = a + d and determinant = a*d - b*c. The quadratic formula gives the two roots.
When are the eigenvalues complex?
The eigenvalues are complex when the discriminant, trace squared minus four times the determinant, is negative. In that case the two eigenvalues are complex conjugates with a real part of trace over two and an imaginary part from the square root of the negated discriminant.
What do the trace and determinant tell me?
The trace equals the sum of the two eigenvalues and the determinant equals their product. These two relationships let you check your answer: add the eigenvalues to recover the trace and multiply them to recover the determinant.
What does a repeated eigenvalue mean?
A repeated eigenvalue occurs when the discriminant is exactly zero, giving one value of multiplicity two. The matrix may still have two independent eigenvectors (if it is a scalar multiple of the identity) or only one, in which case it is defective.
Official sources
- NIST Digital Library of Mathematical Functions: Determinants and linear operators.
- NIST Digital Library of Mathematical Functions: Zeros of polynomials.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.