Simplifying Matrix Powers

Linear Algebra (Matrices) · Axiom Academy

Multiplying A by itself a hundred times is brutal — unless you find the directions where A only stretches. That's the whole reason eigenvalues exist. One vector. A hundred multiplications. No sweat. Raising a matrix to a high power, , looks like a nightmare — every single multiply mixes all the entries together. But hidden inside almost every matrix are a few special directions where A does nothing complicated at all: it just stretches . Watch one of those directions first, then you'll drive the rest yourself. Here , and is one of its special directions — an eigenvector . Press play and apply A over and over. The arrow never leaves its line; it only grows. Each step multiplies its length by exactly , so after n steps it is 3^n times as long: . An eigenvector turns a matrix multiply into a single multiplication by a number: . So why is the eigen-direction such a gift? Switch the starting vector and step n up. The eigenvector stays locked on its line — you always know exactly where it is. A generic vector has no such luck: it swings direction at every step, tumbling toward the eigen-line. To track it you'd have to redo the whole multiply each time. That is the difference between the shortcut and the brute-force mess. On an eigen-line the direction is frozen — only the length changes. Off it, the direction itself is a moving target. Every direction at once: A^n = P D^n P^ -1

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.