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Spectral Radius of a Matrix

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.

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Spectral Radius of a Matrix -

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Spectral Radius of a Matrix

Spectral Radius Calculator

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Eigenvalues Calculator

Answers to Questions (FAQ)

What is the spectral radius of a matrix? (Definition)

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)

How to calculate the spectral radius of a matrix?

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.

How to calculate the eigenvectors of a matrix?

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).

What is a matrix spectrum?

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, https://www.dcode.fr/matrix-spectral-radius

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