Tool for calculating the spectral radius of a matrix, that is to say the maximum value among the absolute values of the eigenvalues of the matrix.

Spectral Radius of a Matrix - dCode

Tag(s) : Matrix

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The spectral radius of a matrix $ M $, denoted $ \rho(M) $, is the highest eigenvalue $ \lambda_{i} $ of the matrix, calculated with absolute value.

$$ \rho(M) = \max \left| \lambda_{i} \right| $$

The spectral radius of a matrix is always positive (thanks to absolute value)

To determine the spectral radius of a matrix, calculate its eigenvalues, then their absolute values, then select the one with the maximum value.

__Example:__ From the 2x2 matrix (order 2) $ M=\begin{bmatrix} 1 & 2 \\ 0 & -3 \end{bmatrix} $, the calculation of eigenvalues gives $ \lambda_1 = -3 $ and $ \lambda_2 = 1 $. The spectral radius is $ \max | \lambda_{i} | = 3 $

dCode has a page dedicated to the calculation of matrix eigenvalues.

Use the matrix eigenvalue calculation page, which contains all the explanations to perform the calculation for all matrix sizes (2x2, 3x3, 4x4, up to NxN).

The spectrum of a matrix is the name sometimes given to the set of its eigenvalues.

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*Spectral Radius of a Matrix* on dCode.fr [online website], retrieved on 2024-06-16,

spectral,radius,eigenvalue,eigen,value,matrix,absolute

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