Finding an Eigenvector

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Finding an Eigenvector Given a matrix and one of its eigenvalues, find a matching eigenvector with row reduction and the null space. The matrix below has as an eigenvalue. Find an eigenvector for — that is, a nonzero vector satisfying . Nicely done. You found an eigenvector by solving a homogeneous system — here is the recipe you just used. Form : Subtract the eigenvalue from each diagonal entry of A ; off-diagonal entries are unchanged. Solve : The solutions of this homogeneous system are exactly the eigenspace. Row reduce to find the null space. Read the null space: A free variable becomes a parameter; write the general solution as a multiple of a basis vector. Any nonzero multiple works: Every scalar multiple of is also an eigenvector, so pick t = 1 for the simplest one. Always verify: Check . Here . ✓︎ The same process works for any eigenvalue. With a larger matrix you may get more than one free variable, giving an eigenspace of dimension greater than one — each basis vector is then a linearly independent eigenvector.

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