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EigenCalc
Enter a 2×2 matrix — get its eigenvalues from the characteristic equation.
Eigenvalues λ of a matrix A satisfy Av = λv — they're the factors by which the matrix stretches its eigenvectors. For a 2×2 matrix, λ = (tr ± √(tr² − 4det)) / 2, where tr = a+d is the trace and det = ad−bc. If the discriminant is negative, eigenvalues are complex (rotation without a real stretch direction).
Eigenvalues determine matrix behavior: all |λ| < 1 means the system decays; any |λ| > 1 means it grows. In PCA, eigenvalues of the covariance matrix rank dimensions by variance. In structural engineering, eigenvalues give natural vibration frequencies. Google's PageRank is the dominant eigenvector of the web graph's transition matrix.
2x2 matrix eigenvalues
Eigenvalue 1
5
Eigenvalue 2
2
What you entered
Trace: tr(A) = a + d
4 + 3= 7Determinant: det(A) = ad − bc
4×3 − 1×2= 10Discriminant: tr² − 4·det
7² − 4×10= 9λ₁ = (tr + √discriminant) / 2
(7 + √9) / 2= 5λ₂ = (tr − √discriminant) / 2
(7 − √9) / 2= 2Result
Eigenvalues: 5, 2
The eigenvalues are 5 and 2 — these are the scalars λ for which Av = λv has a non-trivial solution. The trace (sum of eigenvalues) is 7 and the determinant (product of eigenvalues) is 10.