Math and science::Algebra

Effect of add-identity to eigenvectors

Let $A$ be a matrix that decomposes into:

A = X Λ X^{- 1}

Then it's also true that:

A + β I = X (A + β I) X^{- 1}

or expressed differently, if $C = A + β I$ , then what are the eigenvectors and eigenvalues of $C$ ?

In other words, the eigenvectors of a matrix don't change when any multiple of $I$ is added to the matrix. The eigenvalues do change.

Intuition

To see this in 2D, imagine the columns of a matrix $A$ as vectors, and regress them along the X and Y axes until the matrix is singular. The eigenvectors are the relative ratios of the vectors when in the singular configuration. This configuration doesn't change no matter how many $I$ are added or subtracted from the original matrix $A$ , only the number of $I$ s needed to be subtracted to reach this configuration changes.