Suppose I have a symmetric, positive semidefinite matrix A of dimension n x n. The rank of A is d < n, and I want to find a set of indices of length d such that A[indices, indices] has rank d. For example, if
A <- structure(c(3, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), .Dim = c(4L, 4L))
then I would like to identify one of the sets of indices c(1, 2, 3) or c(1, 2, 4).
I can do this by brute force going through all combinations of rows and columns, but my guess is that there exist much more elegant and efficient methods for doing this.
You can use the QR decomposition, qr
q <- qr(A)
q$pivot[seq(q$rank)]
# [1] 1 2 3
A[,q$pivot[seq(q$rank)]]
# [,1] [,2] [,3]
# [1,] 3 2 1
# [2,] 2 2 1
# [3,] 1 1 1
# [4,] 1 1 1
These resulting columns should be linearly independent.
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