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Math and science::Algebra

Effect of add-identity to eigenvectors

Let \( A \) be a matrix that decomposes into:

\[ A = X\Lambda X^{-1} \]

Then it's also true that:

\[ A + \beta I = X(A + \beta I)X^{-1} \]

or expressed differently, if \( C = A + \beta I \), 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.