The Characteristic Equation

Linear Algebra (Matrices) · Axiom Academy

LESSON The Characteristic Equation Why is exactly the test for an eigenvalue — the singular-matrix logic, made visible. An eigenvector is a nonzero vector that A merely scales by . Start from that definition and move everything to one side: Watch what does to the plane as varies. The unit square's image is a parallelogram whose signed area is exactly : So a nonzero eigenvector exists precisely when is singular (non-invertible). If it were invertible, we could hit both sides with and force — no eigenvector. 2. The Determinant Hits Zero — the Characteristic Equation “ is singular” has a clean test: a square matrix is invertible iff its determinant is nonzero . So a nonzero eigenvector exists exactly when Treat as a function of and slide across the number line. Where the curve crosses zero, collapses — and an eigenvector switches on. (Here .) You derived the characteristic equation from scratch — and saw why a determinant of zero is the same thing as an eigenvector appearing. Scroll up to revisit any step.

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