Diagonalization Theory

Linear Algebra (Matrices) · Axiom Academy

LESSON Diagonalization: A = PDP^ -1 In the eigenvector basis, a matrix stops mixing directions and becomes pure scaling — that change of perspective is diagonalization. 1. A = PDP^ -1 Is a Change of Basis Read A = PDP^ -1 right-to-left as three moves on a vector . First P^ -1 rewrites in the eigenvector coordinates . There, D just scales each coordinate by its eigenvalue. Then P translates back to the standard grid. The round trip does exactly what A does directly. P = eigenvectors in its columns · D = eigenvalues on the diagonal 2. In the Eigenbasis, A Just Scales An eigenvector is a direction A does not turn — it only stretches it: . So if you use the eigenvectors themselves as your axes, A acts on each axis independently , with no cross-talk. Watch A stretch the axis by 3 and leave the axis untouched. . The vector grows along its own line — same direction, tripled length. . Eigenvalue 1 means this direction is fixed — A does nothing to it. Why A = PDP^ -1 follows: stacking for every i gives AP = PD . Since the eigenvectors are independent, P is invertible, so A = PDP^ -1 . The factorization needs n linearly independent eigenvectors to fill the columns of P . A defective matrix has a repeated eigenvalue but too few independent eigenvectors, so no such P exists. Watch : every off-axis vector gets sheared and never settles onto an eigen-direction. n independent eigenvectors — enough directions to fill P and read A off as D .

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