Kernel or Nullspace
Kernel of the linear map defined by matrix M.
nullspace(M) null(M)
M |
Numeric matrix; vectors will be considered as column vectors. |
The kernel (aka null space/nullspace) of a matrix M is the set of
all vectors x for which Ax=0. It is computed from the
QR-decomposition of the matrix.
null is simply an alias for nullspace – and the Matlab name.
If M is an n-by-m (operating from left on
m-dimensional column vectors), then N=nullspace(M) is a
m-by-k matrix whose columns define a (linearly independent)
basis of the k-dimensional kernel in R^m.
If the kernel is only the null vector (0 0 ... 0), then NULL will
be returned.
As the rank of a matrix is also the dimension of its image, the following relation is true:
m = dim(nullspace(M)) + rank(M)
The image of M can be retrieved from orth().
Trefethen, L. N., and D. Bau III. (1997). Numerical Linear Algebra. SIAM, Philadelphia.
M <- matrix(1:12, 3, 4) Rank(M) #=> 2 N <- nullspace(M) # [,1] [,2] [,3] # [1,] 0.4082483 -0.8164966 0.4082483 M M1 <- matrix(1:6, 2, 3) # of rank 2 M2 <- t(M1) nullspace(M1) # corresponds to 1 -2 1 nullspace(M2) # NULL, i.e. 0 0 M <- magic(5) Rank(M) #=> 5 nullspace(M) #=> NULL, i.e. 0 0 0 0 0
Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.