From Eigenvalues to Eigenvectors

Linear Algebra (Matrices) · Axiom Academy

LESSON From Eigenvalues to Eigenvectors You already have the eigenvalues. For each one, solving hands you the eigenvector — the direction A leaves on its own line. 1. Why an Eigenvalue Forces a Direction Take , whose eigenvalues are and . The defining equation rearranges to — so the eigenvector is whatever nonzero the matrix sends to zero. Subtract from the diagonal — the determinant drops to zero A singular matrix has a whole line it sends to 2. Row-Reduce to Read the Eigenvector Knowing a direction is killed isn't enough — you need its components. So solve by row-reducing A-3I . Because the matrix is singular, a row collapses to zero, leaving a free variable — and that free variable is exactly the freedom to scale the eigenvector. Subtract 3 down the diagonal: . Scale row 1, subtract it from row 2 — the bottom row becomes all zeros. v_1-v_2=0 , so v_1=v_2 . One free variable: a line of solutions. Pick v_2=1 : the eigenvector is . 3. Each Eigenvalue, Its Own Eigenvector The procedure is the same for every eigenvalue: substitute it, watch go singular, row-reduce, read the null space. Do it for and a different direction survives — the second eigenvector. Each owns a line (its eigenspace ) that A stretches by exactly that . : row-reduces to v_1+v_2=0 , giving the eigenvector . Since , A leaves this direction's length unchanged. : the eigenvector is stretched three times longer. Two eigenvalues, two perpendicular eigenspace lines, two stretch factors.

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