For tridiagonals T replace T with LDLt

The same number of parameters determine a tridiagonal matrix T and its triangular factors L, D and U. The mapping T→LDU is not well defined for all tridiagonals but, in finite precision arithmetic, L, D and U determine the entries of T to more than working precision. For the solution of linear equations LDUx=b the advantages of factorization are clear. Recent work has shown that LDU is also preferable for the eigenproblem, particularly in the symmetric case. This essay describes two of the ideas needed to compute eigenvectors that are orthogonal without recourse to the Gram–Schmidt procedure when some of the eigenvalues are tightly clustered. In the symmetric case we must replace T, or a translate of T, by its triangular factors LDLt.