Proper use of the rod in the Cholesky decomposition of a positive semidefinite matrix

Question

I do not understand how to use the chol function in R to determine a positive semidefinite matrix. (Or I, and there is an error.) The documentation reads:

If pivot = TRUE, then we can calculate the Cholesky decomposition of a positive semidefinite x. The rank x is returned as attr (Q, "rank"), taking into account numerical errors. The top returns as attr (Q, "pivot"). It is no longer the case that t (Q)% *% Q is equal to x. However, setting the hinge <- attr (Q, "rotation") and oo (axis of rotation), it is true that t (Q [, oo])% *% Q [, oo] is equal to x ...

The following example seems to contradict this description.

 > x <- matrix(1, nrow=3, ncol=3) > Q <- chol(x, pivot=TRUE) > oo <- order(attr(Q, 'pivot')) > t(Q[, oo]) %*% Q[, oo] [,1] [,2] [,3] [1,] 1 1 1 [2,] 1 1 1 [3,] 1 1 3 

The result is not x . Am I using an anchor point incorrectly?

Answer 1

To enter the full rank, i.e. positively defined matrix x , we need

 Q <- chol(x, TRUE) oo <- order(attr(Q, 'pivot')) unpivQ <- Q[, oo] all.equal(crossprod(unpivQ), x) 

For an acceptable rank deficit of input, i.e. positive semidefinite matrix x (an indefinite matrix with negative eigenvalues ​​is illegal, but not checked in chol ), do not forget the zero defective final diagonal block:

 Q <- chol(x, TRUE) r <- attr(Q, 'rank') if (r < nrow(x)) Q[(r+1):nrow(x), (r+1):nrow(x)] <- 0 oo <- order(attr(Q, 'pivot')) unpivQ <- Q[, oo] all.equal(crossprod(unpivQ), x) 

Some people call this the “error” chol , but it’s actually a function of the basic LAPACK dpstrf procedure. Factorization continues until the first diagonal element, which is below the tolerance, leaving the back matrix just untouched upon exit.

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Thanks to Jan for the following observation:

You can use negative indexing R in Q[-(1:r): -(1:r)] <- 0 to skip the if .