spectral_radius

spectral_radius

spectral_radius(X::AbstractMatrix) -> Number

Computes the spectral radius of a matrix X.

Description

The spectral radius of a matrix is defined as the largest absolute eigenvalue of the matrix.

Mathematically, for a given square matrix X, the spectral radius is computed as:

\[ \rho(X) = \max |\lambda_i| \]

where \lambda_i are the eigenvalues of X.

Arguments

• X::AbstractMatrix: A square matrix.

Returns

• The spectral radius of X.

Ex```{julia}le

X = [0.5 0.2; 0.1 0.9]
r = spectral_radius(X)

spectral_radius

spectral_radius(model::Union{VAR,SVAR}) -> Real

Computes the spectral radius of the companion matrix of a fitted VAR model.

The spectral radius of a matrix \(A\) is defined as the largest absolute value among its eigenvalues:

\[ \rho(A) = \max_i |\lambda_i| \]

where \(\lambda_i\) are the eigenvalues of \(A\).

In the context of VAR models, the spectral radius of the companion matrix is a key measure of stability. A VAR model is stable if the spectral radius is strictly less than 1.

Arguments

• model::VAR: A fitted VAR model