Factorizing complex symmetric matrices with positive definite real and imaginary parts

Complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for LU factorization. This result adds to the classes of matrix for which it is known to be safe not to pivot in LU factorization. Block $\mathrm{LDL^T}$ factorization with the pivoting strategy of Bunch and Kaufman is also considered, and it is shown that for such matrices only $1\times 1$ pivots are used and the same growth factor bound of 2 holds, but that interchanges that destroy band structure may be made. The latter results hold whether the pivoting strategy uses the usual absolute value or the modification employed in LINPACK and LAPACK.