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Lotkin Matrix

A Lotkin matrix is a Hilbert matrix with its first row replaced by ones (Lotkin 1955; Higham 1995, p. 51). For an n×n matrix, its entries are

a_(ij)={1 if i=1; 1/(i+j-1) if i>1
(1)

for 1<=i,j<=n.

TestMatrixTableLotkin

Lotkin matrices provide ill-conditioned examples for testing numerical algorithms. The plot compares the singular values of the order-12 Lotkin matrix (blue) and the order-12 Hilbert matrix (orange), in decreasing order. Small singular values indicate sensitivity when solving a linear system of equations.

See also

Hilbert Matrix, Ill-Conditioned Matrix, Test Matrix

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References

Higham, N. J. The Test Matrix Toolbox for Matlab (Version 3.0). Numerical Analysis Report 276. Manchester, England: Manchester Centre for Computational Mathematics, 1995. https://nhigham.com/wp-content/uploads/2023/10/high95m.pdf.Lotkin, M. "A Set of Test Matrices." Math. Tables Aids Comput. 9, 153-161, 1955.

Cite this as:

Weisstein, Eric W. "Lotkin Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LotkinMatrix.html

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