Eigenvector Definition

Linear Algebra (Matrices) · Axiom Academy

LESSON Defining the Eigenvector The one equation — when a matrix sends a vector straight back along its own line. Apply a matrix A to a vector and watch where it lands. For almost every vector, points somewhere new — off the line it started on. An eigenvector is the exception: stays on the same line through the origin, only scaled. The whole definition lives in one equation. Take and . Watch the two sides of get built as separate arrows — and land on top of each other. Right side: scale by — same arrow Subtract the right side from the left to get the equivalent form every eigenvalue method starts from: holds for every , so the zero vector is excluded — it carries no direction to preserve. An eigenvalue may be 0 , negative, or even complex. means ; flips the vector to the opposite ray. and have to live in the same space, so A is . A nonzero solution exists only when is singular — the door to the characteristic equation . You can now state exactly what makes a vector an eigenvector: a matrix sends it straight back along its own line, scaled by its eigenvalue. Scroll up to revisit any step.

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