Wilkinson matrix

{{Short description|Numerical lineral algebra}}

{{One source|date=August 2021}}

In linear algebra, Wilkinson matrices are symmetric, tridiagonal, order-N matrices with pairs of nearly, but not exactly, equal eigenvalues.{{cite book | title = The Algebraic Eigenvalue Problem | author = Wilkinson | publisher = Oxford University Press | year = 1965 | isbn = 0-19-853418-3}} It is named after the British mathematician James H. Wilkinson. For N = 7, the Wilkinson matrix is given by

:$\backslash begin\{bmatrix\}$

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.