fullRank function

Remove Columns (or Rows) From a Matrix to Make It Full Rank

From the QR decomposition with pivoting, (qr(x, tol) if $n \ge p$), if the matrix is not of full rank, the corresponding columns ($n >= p$) or rows ($n < p$) are omitted to form a full rank matrix.

fullRank(x, tol = 1e-7, qrx = qr(x, tol=tol))

Arguments

• x: a numeric matrix of dimension $n * p$, or a similar object for which qr() works.
• tol: tolerance for determining rank (deficiency). Currently is simply passed to qr.
• qrx: optionally may be used to pass a qr(x, ..); only used when p <= n.

Returns

a version of the matrix x, with less columns or rows if x's rank was smaller than min(n,p).

If x is of full rank, it is returned unchanged.

Author(s)

Martin Maechler

Note

This is useful for robustness algorithms that rely on $X$ matrices of full rank, e.g., adjOutlyingness.

This also works for numeric data frames and whenever qr() works correctly.

See Also

qr; for more sophisticated rank determination, rankMatrix from package list("Matrix").

Examples

stopifnot(identical(fullRank(wood), wood))

## More sophisticated and delicate
dim(T <- tcrossprod(data.matrix(toxicity))) # 38 x 38
dim(T. <- fullRank(T)) # 38 x 10
if(requireNamespace("Matrix")) {
  rMmeths <- eval(formals(Matrix::rankMatrix)$method)
  rT. <- sapply(rMmeths, function(.m.) Matrix::rankMatrix(T., method = .m.))
  print(rT.) # "qr" (= "qrLinpack"): 13,  others rather 10
}
dim(T.2 <- fullRank(T, tol = 1e-15))# 38 x 18
dim(T.3 <- fullRank(T, tol = 1e-12))# 38 x 13
dim(T.3 <- fullRank(T, tol = 1e-10))# 38 x 13
dim(T.3 <- fullRank(T, tol = 1e-8 ))# 38 x 12
dim(T.) # default from above          38 x 10
dim(T.3 <- fullRank(T, tol = 1e-5 ))# 38 x 10 -- still

plot(svd(T, 0,0)$d, log="y", main = "singular values of T", yaxt="n")
axis(2, at=10^(-14:5), las=1)
## pretty clearly indicates that  rank 10  is "correct" here.