nullspace | R Documentation |
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.
Rank
, orth
, MASS::Null
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
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