Spectral Radius - Theorem (Gelfand's Formula, 1941)

Theorem (Gelfand's Formula, 1941)

For any matrix norm ||·||, we have

In other words, Gelfand's formula shows how the spectral radius of A gives the asymptotic growth rate of the norm of Ak:

for

Proof: For any ε > 0, consider the matrix

Then, obviously,

and, by the previous theorem,

That means, by the sequence limit definition, a natural number N₁ ∈ N exists such that

which in turn means:

or

Let's now consider the matrix

Then, obviously,

and so, by the previous theorem, is not bounded.

This means a natural number N₂ ∈ N exists such that

which in turn means:

or

Taking

and putting it all together, we obtain:

which, by definition, is

Gelfand's formula leads directly to a bound on the spectral radius of a product of finitely many matrices, namely assuming that they all commute we obtain

Actually, in case the norm is consistent, the proof shows more than the thesis; in fact, using the previous lemma, we can replace in the limit definition the left lower bound with the spectral radius itself and write more precisely:

which, by definition, is

Example: Let's consider the matrix

whose eigenvalues are 5, 10, 10; by definition, its spectral radius is ρ(A)=10. In the following table, the values of for the four most used norms are listed versus several increasing values of k (note that, due to the particular form of this matrix,):