Eigenspaces

Linear Algebra (Matrices) · Axiom Academy

One eigenvalue doesn't give you one eigenvector — it gives you a whole subspace of them. 1. A Whole Line of Eigenvectors Take with eigenvalue . Watch the line y=x : every vector lying on it is stretched by exactly 3 and stays on the line. A vector pointing off the line gets knocked into a new direction — it is not just scaled. 2. The Eigenspace Is a Null Space Rewrite as . So the eigenvectors for — plus — are precisely the vectors that sends to the origin: its null space . The animation builds A-3I , then shows the eigen-line is closed under addition and scaling, which is what makes it a subspace. 3. Dimension Is the Geometric Multiplicity How big can an eigenspace get? Its dimension is the geometric multiplicity of . It can be a line (dimension 1) or a whole plane (dimension 2), and it is always pinned between 1 and the eigenvalue's algebraic multiplicity. Watch three eigenspaces for — a line, a plane, and a "defective" line — fill out their dimensions. You've seen the eigenspace as the full set of eigenvectors for — a subspace whose size measures eigenvector freedom. Scroll up to revisit any step.

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