Complex eigenvalues
Some real matrices have no real eigenvectors, because their characteristic equation has complex roots such as i and -i. Rotations do this.
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Analogy
Spinning a wheel: no straight line through the centre keeps its direction.
Example
A = [[0,-1],[1,0]].det(A - L I) = L^2 + 1.L^2 + 1 = 0 gives L = i or -i.No real eigenvector exists.
In practice
Complex eigenvalues signal rotation or oscillation in a system.
Questions people ask
Why does a rotation have no real eigenvectors?
A real eigenvector would stay on the same line after the transformation. A 90 degree rotation turns every non-zero vector off its line, so none qualifies. The algebra agrees: lambda squared plus 1 equals 0 has no real solutions. Its solutions, i and minus i, are complex.
What do complex eigenvalues tell me about a system?
They show that the system turns or oscillates instead of simply growing or shrinking. The real part controls growth or decay, and the imaginary part controls the speed of rotation. Examples include springs, circuits and predator-prey cycles, where numbers rise and fall in turn.