spectral radius

Spectral Radius

The spectral radius of a square matrix $A$, denoted $\rho (A)$, is the maximum absolute value of its eigenvalues:

$$\rho (A)=\max \{|{\lambda }_1|,|{\lambda }_2|,\dots ,|{\lambda }_n|\}$$

where ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ are the eigenvalues of $A$.

The spectral radius provides a measure of how much the matrix $A$ can amplify vectors in the long run when applied repeatedly.

This makes it critical for understanding the stability of dynamical systems and iterative processes. In a linear dynamical system $x_{t+1}=A{x}_t$, the spectral radius determines whether the system converges, oscillates, or diverges as $t\to \infty$.

Note

For any matrix norm $\Vert \cdot \Vert$, the spectral radius satisfies $\rho (A)\le \Vert A\Vert$. Additionally, for any $\epsilon >0$, there exists a matrix norm $\Vert \cdot {\Vert }_{\epsilon }$ such that $\Vert A{\Vert }_{\epsilon }<\rho (A)+\epsilon$. This means the spectral radius is the infimum of all matrix norms of $A$.

Example

Consider the matrix $A=\left[\begin{array}{cc}3& 1\\ 0& 2\end{array}\right]$. Its eigenvalues are ${\lambda }_1=3$ and ${\lambda }_2=2$. Therefore, the spectral radius is $\rho (A)=\max \{|3|,|2|\}=3$.