Wilkinson matrix

In linear algebra, Wilkinson matrices are symmetric, tridiagonal, order-N matrices with pairs of nearly, but not exactly, equal eigenvalues.^[1] It is named after the British mathematician James H. Wilkinson. For N = 7, the Wilkinson matrix is given by

$\begin{bmatrix} 3 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 2 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 3 \\ \end{bmatrix}.$

Wilkinson matrices have applications in many fields, including scientific computing, numerical linear algebra, and signal processing.

References

1. Wilkinson (1965). The Algebraic Eigenvalue Problem. Oxford University Press. ISBN 0-19-853418-3.