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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:

[\[ A + \beta I = \quad ? \]]

or expressed differently, if \( C = A + \beta I \), then what are the eigenvectors and eigenvalues of \( C \)?