Matrix Neumann Series

Concept Overview

The Neumann series for matrices is a generalization of the geometric series for real numbers. For a matrix A, the series is defined as the sum of its non-negative powers: S = I + A + A² + ... + A^k + ... If the series converges, its limit is the inverse matrix (I - A)^-1. This theory is essential for solving linear systems and understanding iterative methods.

Worked Example

Consider a diagonal matrix A where r = 0.5:

A = [ 0.5  0.0 ]
    [ 0.0 -0.5 ]

S_3 = I + A + A^2 + A^3

I = [ 1.0  0.0 ]
    [ 0.0  1.0 ]

A = [ 0.5  0.0 ]
    [ 0.0 -0.5 ]

A^2 = [ 0.25 0.00 ]
      [ 0.00 0.25 ]

A^3 = [ 0.125  0.000 ]
      [ 0.000 -0.125 ]

S_3 = [ 1.875  0.000 ]
      [ 0.000  0.625 ]

(I - A)^-1 = [ 1-0.5  0.0  ]^-1  = [ 0.5  0.0 ]^-1 = [ 2.0      0.0 ]
             [ 0.0   1-(-0.5) ]      [ 0.0  1.5 ]      [ 0.0  0.666... ]

Experiment

Follow these steps to observe the convergence behavior of the series.

Change the Scale to 1.5 in the Matrix Neumann Series module.
Observe that while the matrix (I - A) is still invertible, the series diverges because the spectral radius is greater than 1.
Change the Family to Nilpotent with a Scale of 2.
Observe that despite the matrix norm being greater than 1, the series converges exactly in finite steps because the matrix is nilpotent (A² = 0).

Limitations

The primary limitation of using the Neumann series directly for computation is that it requires the spectral radius of the matrix to be strictly less than 1 to guarantee convergence. In practical applications, assessing the spectral radius can be computationally expensive. Additionally, when the spectral radius is close to 1, the series may converge very slowly, making it inefficient compared to direct methods like LU decomposition.

References

For further reading on this topic, consult the following resources:

What is the Spectral Radius of a Matrix? - N. Higham

Matrix Neumann Series