Rank 1 and rank 2 cholesky updates?

General Usage
,
2

A condition to have Cholesky decompositions is that the matrix is positive-definite which requires in particular: symmetric (real case) or Hermitian (complex case). The matrix A + e_i e_j^T would break this assumption.
However, adding A + e_i e_j^T + e_j e_i^T stays at least symmetric/Hermitian. (But might fail to be positive-definite.)

You could use three steps:
A + (e_i + e_j) (e_i + e_j)^T - e_i e_i^T - e_j e_j^T

That should correspond to one low-rank update and two downgrades, kind of like

a = randn(3,3)
A = a' * a
ei = [1.0, 0, 0]; ej = [0.0, 1, 0]
C = cholesky(A)
lowrankupdate!(C, ei + ej)
lowrankdowndate!(C, ei)
lowrankdowndate!(C, ej)

However, this will fail if a non-positive-definite matrix occurs as any of the results.

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