---
tag: 3KX0KQ7V05
kind: statement
role: theorem
name: Gelfand's formula
msc: [15A18, 15A60]
uses: [9V7YQA85PC, 54T6Y6DSY4, 05M5SWSZZD]
about: [9V7YQA85PC]
sources:
  - url: https://en.wikipedia.org/wiki/Spectral_radius
    locator: Gelfand's formula
---

## Statement

Let $n \ge 1$, let $A$ be an $n \times n$ complex matrix, and let $\norm{\cdot}$ be a [[matrix norm|54T6Y6DSY4]] on $\C^{n \times n}$.
Then the [[spectral radius|9V7YQA85PC]] of $A$ is the [[limit|05M5SWSZZD]]
$$\rho(A) = \lim_{k \to \infty} \norm{A^k}^{1/k}.$$

## Remarks

For a submultiplicative norm, $\rho(A)^k = \rho(A^k) \le \norm{A^k}$, which gives $\rho(A) \le \norm{A^k}^{1/k}$ for every $k$.
For $\varepsilon > 0$, the matrix $A / (\rho(A) + \varepsilon)$ has spectral radius less than $1$, so its powers tend to $0$ by the Jordan normal form; this gives $\norm{A^k} \le (\rho(A) + \varepsilon)^k$ for large $k$.
Any two norms are equivalent, and $c^{1/k} \to 1$ for $c > 0$, so the limit does not depend on the norm.
Example: $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ has $A^k = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$, and with the maximum row sum norm $\norm{A^k}^{1/k} = (1 + k)^{1/k} \to 1 = \rho(A)$, although $\norm{A} = 2$.
The same formula holds for elements of a complex Banach algebra; see the [[spectral radius formula|KX2XY7S3X6]].

## Explanations

### L1 human

The spectral radius is the largest absolute value of an eigenvalue.
A norm of $A$ can overestimate it, but the norms of the powers $A^k$ grow like $\rho(A)^k$.
Taking the $k$-th root and letting $k$ grow recovers $\rho(A)$ exactly, whatever norm is used.

### L2 human

The formula explains why $A^k \to 0$ if and only if $\rho(A) < 1$, which is the convergence criterion for iterative methods such as Jacobi and Gauss–Seidel.
It also shows that $\rho(A) \le \norm{A}$ for every submultiplicative norm, and that for every $\varepsilon > 0$ there is an operator norm with $\norm{A} \le \rho(A) + \varepsilon$.
In the Banach algebra setting, the same statement is proved with the resolvent and complex analysis instead of the Jordan form.