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(·). Let λ1, ..., λn be the (real or complex) eigenvalues of a matrix A ∈ Cn×n. Then its spectral radius ρ(A) is defined as: The condition number of A {displaystyle A} can be expressed using the spectral radius as ρ ( A ) ρ ( A − 1 ) {displaystyle ho (A) ho (A^{-1})} . The spectral radius is a sort of infimum of all norms of a matrix. On the one hand, ρ ( A ) ⩽ ‖ A ‖ {displaystyle ho (A)leqslant |A|} for every natural matrix norm ‖ ⋅ ‖ {displaystyle |cdot |} , and on the other hand, Gelfand's formula states that ρ ( A ) = lim k → ∞ ‖ A k ‖ 1 / k {displaystyle ho (A)=lim _{k o infty }|A^{k}|^{1/k}} ; both these results are shown below. However, the spectral radius does not necessarily satisfy ‖ A v ‖ ⩽ ρ ( A ) ‖ v ‖ {displaystyle |Amathbf {v} |leqslant ho (A)|mathbf {v} |} for arbitrary vectors v ∈ C n {displaystyle mathbf {v} in mathbb {C} ^{n}} . To see why, let r > 1 {displaystyle r>1} be arbitrary and consider the matrix C r = ( 0 r − 1 r 0 ) {displaystyle C_{r}={egin{pmatrix}0&r^{-1}\r&0end{pmatrix}}} . The characteristic polynomial of C r {displaystyle C_{r}} is λ 2 − 1 {displaystyle lambda ^{2}-1} , hence its eigenvalues are ± 1 {displaystyle pm 1} , and thus ρ ( C r ) = 1 {displaystyle ho (C_{r})=1} . However C r e 1 = r e 2 {displaystyle C_{r}mathbf {e} _{1}=rmathbf {e} _{2}} , so ‖ C r e 1 ‖ = r > 1 = ρ ( C r ) ‖ e 1 ‖ {displaystyle |C_{r}mathbf {e} _{1}|=r>1= ho (C_{r})|mathbf {e} _{1}|} for ‖ ⋅ ‖ {displaystyle |cdot |} being any ℓ p {displaystyle ell ^{p}} norm on C n {displaystyle mathbb {C} ^{n}} . What still allows ‖ C r k ‖ 1 / k → 1 {displaystyle |C_{r}^{k}|^{1/k} o 1} as k → ∞ {displaystyle k o infty } is that C r 2 = I {displaystyle C_{r}^{2}=I} , making ‖ C r k ‖ 1 / k ⩽ ‖ C r ‖ 1 / k = r 1 / k → 1 {displaystyle |C_{r}^{k}|^{1/k}leqslant |C_{r}|^{1/k}=r^{1/k} o 1} as k → ∞ {displaystyle k o infty } . does hold when A {displaystyle A} is a Hermitian matrix and ‖ ⋅ ‖ {displaystyle |cdot |} is the Euclidean norm. 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 C such that the degree of every vertex of the graph is smaller than C). In this case, for the graph G define: Let γ be the adjacency operator of G: