numericalRank -- numerical rank of a matrix

Synopsis

• Usage:

  r = numericalRank M

  B = isFullNumericalRank M

• Inputs:
  • M, a matrix, a matrix with real or complex entries
• Optional inputs:
  • Threshold => ..., default value .0001
• Outputs:
  • r, an integer
  • B, a Boolean value

Description

numericalRank finds an approximate rank of the matrix M.

isFullNumericalRank = M is _not_ rank-deficient.

Let \sigma_1,...,\sigma_n be the singular values of M.

If Threshold is >1, then to establish numerical rank we look for the first large gap between two consecutive singular values. The gap between \sigma_i and \sigma_{i+1} is large if \sigma_i/\sigma_{i+1} > Threshold.

If Threshold is <=1, then the rank equals the number of singular values larger then Threshold.

i1 : options numericalRank

o1 = OptionTable{Threshold => .0001}

o1 : OptionTable

i2 : numericalRank matrix {{2,1},{0,0.001}}

o2 = 2

i3 : numericalRank matrix {{2,1},{0,0.0001}}

o3 = 1

Caveat

See also

• SVD -- singular value decomposition of a matrix