Eigenvalue & Eigenvector Calculator (2x2)
Enter the four entries of a 2x2 matrix [[a, b], [c, d]] to calculate its eigenvalues and eigenvectors.
Last updated: September 5, 2026
Frequently asked questions
How do I find eigenvalues of a 2x2 matrix?
Solve det(A − λI) = 0, which for a 2x2 matrix [[a,b],[c,d]] simplifies to λ² − (a+d)λ + (ad−bc) = 0 - a quadratic equation in λ, using the matrix's trace (a+d) and determinant (ad−bc). This calculator applies that formula directly.
How do I find the eigenvector once I have an eigenvalue?
Solve (A − λI)v = 0 for the vector v - for a 2x2 matrix this reduces to one independent equation, since the system is singular by construction. This calculator solves that directly from the b or c entry and the eigenvalue.
What does it mean if the eigenvalues are complex?
Complex eigenvalues mean the matrix has no real eigenvectors - geometrically, this happens for matrices that rotate vectors (like a rotation matrix) rather than just stretching them along fixed directions, since a rotation has no direction that stays unchanged (except a 180° rotation).
What real-world things do eigenvalues and eigenvectors represent?
They describe directions that a matrix transformation doesn't rotate, only stretches or shrinks - the eigenvector is that unchanged direction, and the eigenvalue is the stretch factor. This shows up in applications like vibration analysis, principal component analysis, and Google's PageRank algorithm.
What are the eigenvalues of the matrix [[2,1],[1,2]]?
3 and 1 - trace = 2+2 = 4, determinant = (2×2)-(1×1) = 3, so λ² − 4λ + 3 = 0 factors to (λ−3)(λ−1) = 0, giving λ = 3 and λ = 1.
Can a 2x2 matrix have only one eigenvalue?
Yes - this happens when the discriminant of the characteristic equation is zero (a 'repeated' or 'degenerate' eigenvalue), which occurs for matrices like [[2,1],[0,2]], where both roots of the quadratic coincide at the same value.