A note on the growth factor in Gaussian elimination for generalized Higham matrices

The Higham matrix is a complex symmetric matrix A = B + i C A=B+\mathrm {i}C , where both matrices B B and C C are real, symmetric, and positive definite and i = − 1 \mathrm {i}=\sqrt {-1} is the imaginary unit. For any Higham matrix A A , George, Ikramov, and Kucherov [Numer. Linear Algebra Appl. 9 (2002), pp. 107–114] showed that the growth factor in Gaussian elimination is less than 3 3 . In this paper, based on the previous results, a new bound of the growth factor is obtained by using the maximum of the condition numbers of the (complex) matrices B B and C C for the generalized Higham matrix A A also defined by George, Ikramov, and Kucherov [Numer. Linear Algebra Appl. 9 (2002), pp. 107–114], which strengthens this bound to 2 2 and proves Higham’s conjecture.