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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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Eigenvalues Calculator

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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