Eigenvalues and Eigenvectors Summary

Linear Algebra (Matrices) · Axiom Academy

SUMMARY Eigenvalues and Eigenvectors Unit 5 in one place: special directions, the characteristic equation, eigenspaces, diagonalization, and why eigenvalues run so much of applied math. An eigenvector is a special direction a matrix only scales : . The scale factor is its eigenvalue. Eigenvalues are the roots of the characteristic equation ; their sum is and their product is . Each eigenvector lives in the eigenspace — solve to find it. With n independent eigenvectors, A diagonalizes : A=PDP^ -1 , which makes powers trivial: . Eigenvalues govern long-term behavior — the dominant one decides growth, decay, or oscillation in dynamical systems. Core Concept The Eigen-Equation An eigenvector is a nonzero direction that the transformation leaves on its own line — it gets stretched, shrunk, or flipped by the scalar , never rotated off its span. Reads as: stretches, shrinks, flips, collapses onto the null space. Watch out for: any scalar multiple is also an eigenvector — direction is what matters, not length. Core Concept Characteristic Equation Eigenvalues are exactly the that make singular. Expanding the determinant gives the characteristic polynomial — degree n for an matrix — whose roots are the eigenvalues. Quick checks: roots sum to and multiply to . Watch out for: counting multiplicity (and complex conjugate pairs), an matrix has n eigenvalues. Core Concept Eigenvectors & Eigenspaces

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