Verify an eigenpair without requiring a fixed vector sign

Define the task

An eigenvector can be multiplied by −1 while describing the same direction. A comparison of software outputs should first align eigenvalues, then check whether the vectors agree up to sign. For repeated eigenvalues, different orthonormal bases of the same eigenspace can be valid, so comparing columns one by one can be misleading.

Carry out the check

For a proposed pair (lambda,v), calculate A times v and subtract lambda times v. Check the residual relative to matrix and vector magnitudes. Verify normalization separately. If principal-component scores change sign along with their loading vector, their associated geometry can remain identical. Record any sign convention used for figures so repeated reports remain readable.

Worked hypothetical example

For a hypothetical matrix with rows (2,1) and (1,2), vector (1,1) has eigenvalue 3, and vector (1,−1) has eigenvalue 1. Their negatives satisfy the same equations. The normalized vectors divide each coordinate by the square root of two. A change in sign therefore needs no scientific explanation, while a large equation residual warrants investigating ordering, scaling or numerical implementation.

Method checked 12 November 2026 against R documentation: Spectral Decomposition of a Matrix.