Spectral radius

In mathematics, the spectral radius of a square matrix or a bounded linear operator is the largest absolute value of its eigenvalues (i.e. supremum among the absolute values of the elements in its spectrum). It is sometimes denoted by ρ(·).
Matrices.
Let be the (real or complex) eigenvalues of a matrix . Then its spectral radius is defined as:
The spectral radius is a sort of infimum of all norms of a matrix. Indeed, on the one hand, formula_2 for every natural matrix norm formula_3; and on the other hand, Gelfand's formula states that formula_4. Both these results are shown below. 
However, the spectral radius does not necessarily satisfy formula_5 for arbitrary vectors formula_6. To see why, let formula_7 be arbitrary and consider the matrix
The characteristic polynomial of formula_9 is formula_10, so its eigenvalues are formula_11 and thus formula_12. However, formula_13. As a result, for any formula_14 norm,
As an illustration of Gelfand's formula, note that formula_16 as formula_17, since formula_18 if formula_19 is even and formula_20 if formula_19 is odd.
A special case in which formula_5 for all formula_6 is when formula_24 is a Hermitian matrix and formula_25 is the Euclidean norm. This is because any Hermitian Matrix is diagonalizable by a unitary matrix, and unitary matrices preserve vector length:
Graphs.
The spectral radius of a finite graph is defined to be the spectral radius of its adjacency matrix.
This definition extends to the case of infinite graphs with bounded degrees of vertices (i.e. there exists some real number such that the degree of every vertex of the graph is smaller than ). In this case, for the graph define:
Let be the adjacency operator of :
The spectral radius of is defined to be the spectral radius of the bounded linear operator .
Upper bound.
Upper bounds for spectral radius of a matrix.
The following proposition shows a simple yet useful upper bound for the spectral radius of a matrix:
Proposition. Let with spectral radius and a consistent matrix norm . Then for each integer formula_29:
Proof
Let be an eigenvector-eigenvalue pair for a matrix "A". By the sub-multiplicative property of the matrix norm, we get:
and since we have
and therefore
Upper bounds for spectral radius of a graph.
There are many upper bounds for the spectral radius of a graph in terms of its number "n" of vertices and its number "m" of edges. For instance, if
where formula_35 is an integer, then
Power sequence.
Theorem.
The spectral radius is closely related to the behaviour of the convergence of the power sequence of a matrix; namely, the following theorem holds:
Proof of theorem.
Assume the limit in question is zero, we will show that . Let be an eigenvector-eigenvalue pair for "A". Since we have:
and, since by hypothesis , we must have
which implies |λ| < 1. Since this must be true for any eigenvalue λ, we can conclude ρ("A") < 1.
Now assume the radius of is less than . From the Jordan normal form theorem, we know that for all , there exist with non-singular and block diagonal such that:
with
where
It is easy to see that
and, since is block-diagonal,
Now, a standard result on the -power of an formula_46 Jordan block states that, for formula_47:
Thus, if formula_49 then for all formula_50. Hence for all we have:
which implies
Therefore,
On the other side, if formula_54, there is at least one element in which doesn't remain bounded as k increases, so proving the second part of the statement.
Gelfand's formula.
Theorem.
The next theorem gives the spectral radius as a limit of matrix norms.
Proof.
For any , first we construct the following two matrices:
Then:
First we apply the previous theorem to :
That means, by the sequence limit definition, there exists such that for all "k" ≥ N+,
so
Applying the previous theorem to implies formula_61 is not bounded and there exists such that for all "k" ≥ N−,
so
Let then we have:
which, by definition, is
Gelfand corollaries.
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
where the + means that the limit is approached from above.
Example.
Consider the matrix
whose eigenvalues are ; by definition, . In the following table, the values of formula_70 for the four most used norms are listed versus several increasing values of k (note that, due to the particular form of this matrix,formula_71):
Bounded linear operators.
For a bounded linear operator and the operator norm ||·||, again we have
A bounded operator (on a complex Hilbert space) is called a spectraloid operator if its spectral radius coincides with its numerical radius. An example of such an operator is a normal operator.