For every self-consistent matrix norm, $||\cdot||$ , and every square matrix $\mv{A}$ we can write

Note: $\rho(\mv{A})$ denotes the spectral radius of $\mv{A}$ .

This theorem also generalizes to infinite dimensions and plays an important role in the theory of operator algebras. If $\mathcal{A}$ is a Banach algebra with norm $||\cdot||$ and $A \in \mathcal{A}$ , then we have

It is worth pointing out that the self-consistency condition which was imposed on the matrix norm is part of the definition of a Banach algebra. A common case of the infinite-dimensional generalization occurs when $\mathcal{A}$ is the algebra of bounded operators on a Hilbert space -- the operators may be regarded as an infinite-dimensional generalization of the square matrices.