Diagonalization as Basis Change

Linear Algebra (Matrices) · Axiom Academy

LESSON Diagonalization Is a Change of Basis A "complicated" matrix A = PDP^ -1 is just independent scaling — once you look at it in its own eigenvector basis. 1. The Same Map: Skew Here, Pure Stretch There Watch act on a ring of unit vectors. A generic vector gets turned and stretched — that mixing is what makes A look complicated. But two directions never leave their own line: the eigenvector (1,1) is stretched by , and the eigenvector (1,-1) is fixed by . Along those axes the map is pure scaling . stretched by 3 — stays on its line unchanged ( ) — stays on its line 2. Reading A = PDP^ -1 Right to Left Stack the eigenvectors as the columns of . Then P is the change-of-basis matrix into the standard basis, and P^ -1 takes a vector the other way, into eigen-coordinates. Applying A is the three-step pipeline below — and in the middle, in eigen-coordinates, A is simply the diagonal . Re-express the vector in the eigenvector basis: how much of each eigen-direction it contains. Multiply the first coordinate by and the second by . No mixing. Rebuild the scaled vector in the original coordinates. The net effect equals A . Same output as — but routed through a basis where the work is trivial. 3. Powers Become Trivial: A^n = PD^nP^ -1

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