About the Eigenvalue Calculator

For a 2×2 matrix, eigenvalues come from the characteristic equation λ² − trace·λ + det = 0. The quadratic formula yields both roots exactly: λ = (trace ± √(trace² − 4·det)) ÷ 2. For [[4, 1], [2, 3]]: trace 7, det 10, discriminant 9 → λ = (7 ± 3)/2 = 5 and 2.

What Eigenvalues Tell You

In dynamics, eigenvalues of the system matrix decide stability: all real parts negative means decay to equilibrium; any positive real part means growth; nonzero imaginary parts mean oscillation or spiraling. This is the same math behind vibrating beams, population models, and circuit transients.