The spectral radius is a property of the eigenvalues of a matrix. I'll try to explain it with a minimum of typical mathematics bullshit.

To find the spectral radius, take all the eigenvalues and plot them on the imaginary axis. Then, draw a circle such that the circle contains all the eigenvalues within it, with the eigenvalue farthest from the origin being on the edge of the circle. The radius of that circle is called the spectral radius.

In other words,

φ(M) = max { |λ| , for all eigenvalues of the matrix M }

For those of you who have your character set messed up (like me), that first symbol is phi and the second one is lambda.

So, what use is the spectral radius? It's useful in determining if Jacobi iteration (a method of solving systems of equations) converges. In particular, the spectral radius must be strictly less than 1 for Jacobi iteration to converge. There are other uses, though I have either forgotten them or haven't learned them yet.

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