Let $A$ be a non-negative square matrix. Normalize $A$ by its spectral radius $\sigma(A)$, and call it $A_2 = A/\sigma(A)$. Does this normalization preserve the ratio between the two largest eigenvalues of $A$? That is, is the ratio $\lambda_1/\lambda_2$ the same for $A$ and $A_2$, where $\lambda_1$ and $\lambda_2$ are the absolute values of the first and second largest eigenvalues of the two matrices?
Does spectral normalization preserve eigenvalue ratios?
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linear-algebra
matrices
eigenvalues-eigenvectors