deflation

Once one eigenvalue is known, mathematically removing it so the SAME method can be reused, unmodified, to find the next one.

Deflation builds, from a symmetric matrix $A$ , a known eigenvalue $\lambda_1$ , and its unit eigenvector $\mathbf v_1$ , a new matrix $A_2=A-\lambda_1\mathbf v_1\mathbf v_1^{\mathsf T}$ sharing every eigenvalue with $A$ except $\lambda_1$ , which is replaced by $0$ -- every eigenvector orthogonal to $\mathbf v_1$ (every OTHER eigenvector, for a symmetric $A$ ) is left untouched. Running the power method again on $A_2$ finds $A$ 's second-largest eigenvalue without re-deriving the whole characteristic polynomial.

Examples

Having found $\lambda_1=5$ and its eigenvector, running the power method again on the deflated matrix $A_2$ converges to $A$ 's NEXT-largest eigenvalue -- $\lambda_1$ itself no longer competes.