Spectral radius
Let be the eigenvalues of a matrix . The spectral radius of is defined as
formula_1
The spectral radius can be thought of as an 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 of 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
formula_8. 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,
formula_14 
As an illustration of Gelfand's formula, note that formula_15 as formula_16, since formula_17 if formula_18 is even and formula_19 if formula_18 is odd. A special case in which formula_5 for all formula_6 is when formula_23 is a Hermitian matrix and formula_24 is the Euclidean norm. This is because any Hermitian Matrix is diagonalizable by a unitary matrix, and unitary matrices preserve vector length. As a result,
 formula_25 In the context of a bounded linear operator on a Banach space, the eigenvalues need to be replaced with the elements of the spectrum of the operator, i.e. the values formula_26 for which formula_27 is not bijective. We denote the spectrum by
formula_28 
The spectral radius is then defined as the supremum of the magnitudes of the elements of the spectrum:
formula_29
Gelfand's formula, also known as the spectral radius formula, also holds for bounded linear operators: letting formula_3 denote the operator norm, we have
formula_31
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. 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:
formula_32
Let be the adjacency operator of :
formula_33
The spectral radius of is defined to be the spectral radius of the bounded linear operator . The following proposition gives simple yet useful upper bounds on the spectral radius of a matrix. Proposition. Let with spectral radius and a consistent matrix norm . Then for each integer formula_34:
formula_35
Proof
Let be an eigenvector-eigenvalue pair for a matrix "A". By the sub-multiplicativity of the matrix norm, we get:
formula_36
Since , we have
formula_37
and therefore
formula_35
concluding the proof. 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
formula_39
where formula_40 is an integer, then
formula_41 The spectral radius is closely related to the behavior of the convergence of the power sequence of a matrix; namely as shown by the following theorem. Theorem. Let with spectral radius . Then if and only if
formula_42
On the other hand, if , formula_43. The statement holds for any choice of matrix norm on . Proof
Assume that formula_44 goes to zero as formula_18 goes to infinity. We will show that . Let be an eigenvector-eigenvalue pair for "A". Since , we have
formula_46
Since by hypothesis, we must have
formula_47
which implies |λ| < 1. Since this must be true for any eigenvalue λ, we can conclude that . 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:
formula_48
with
formula_49
where
formula_50
It is easy to see that
formula_51
and, since is block-diagonal,
formula_52
Now, a standard result on the -power of an formula_53 Jordan block states that, for formula_54:
formula_55
Thus, if formula_56 then for all formula_57. Hence for all we have:
formula_58
which implies
formula_59
Therefore,
formula_60
On the other side, if formula_61, there is at least one element in that does not remain bounded as increases, thereby proving the second part of the statement. Gelfand's formula, named after Israel Gelfand, gives the spectral radius as a limit of matrix norms. For any matrix norm we have
formula_62. Moreover, in the case of a consistent matrix norm formula_63 approaches formula_64 from above (indeed, in that case formula_65 for all formula_18). For any , let us define the two following matrices:
formula_67
Thus,
formula_68
We start by applying the previous theorem on limits of power sequences to :
formula_69
This shows the existence of such that, for all ,
formula_70
Therefore,
formula_71
Similarly, the theorem on power sequences implies that formula_72 is not bounded and that there exists such that, for all "k" ≥ N−,
formula_73
Therefore,
formula_74
Let }. Then,
formula_75
that is,
formula_76
This concludes the proof. Gelfand's formula yields a bound on the spectral radius of a product of commuting matrices: if formula_77 are matrices that all commute, then
formula_78 Consider the matrix
formula_79
whose eigenvalues are ; by definition, . In the following table, the values of formula_80 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_81):
Definition
In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectrum. The spectral radius is often denoted by .