Spectral radius

Remarks

Example: $A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ has characteristic polynomial $x^2 + 1$, so $\sigma(A) = \set{i, -i}$ and $\rho(A) = 1$.

The spectral radius is not a norm on matrices. For $N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, the only eigenvalue is $0$, so $\rho(N) = 0$ although $N \ne 0$.

For every operator norm $\norm{\cdot}$, $\rho(A) \le \norm{A}$, and Gelfand's formula gives $\rho(A) = \lim_{k \to \infty} \norm{A^k}^{1/k}$. The powers $A^k$ tend to $0$ if and only if $\rho(A) < 1$. For a normal matrix, $\rho(A)$ equals the operator norm induced by the Euclidean norm. The same definition applies to a linear map of a finite-dimensional complex vector space.