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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Spectral Radius of a Matrix
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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