The spectral radius of a square matrix or a bounded linear operator is the largest absolute value of its spectrum.

Let $A$ be an $n\times n$ matrix. Its spectral radius $\rho(A)$ is the largest absolute value of its eigenvalues $\lambda_i$, i.e. $$ \rho(A) = \max_{1\leq i\leq n} |\lambda_i| $$

A closely related term is the spectral norm. These are not necessarily the same: for instance, if $A=\pmatrix{0 & 1\\ 0 & 0}$, then $\rho(A)=0$ and $||A||_2=1$. In general we have $\rho(A)\leq ||A||_k$ and a powerful result known as Gelfand's formula gives $\rho(A) = \lim_{k\to \infty}||A||_k ^{1/k}$. See spectral-norm for these questions.

Questions about the spectral radius should usually contain some combination of the tags linear-algebra, eigenvalues-eigenvectors, matrices, svd, or similar.

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