What is the result of the operation A\B, where A(1, m) and B (1, m)?
In the manual it is written:
A\B returns a least-squares solution to the system of equations A*x= B.
So it means x = inv (A'*A)*A'*B? However, the matrix A'*A is singular...
Let us suppose:
A=[1 2 3]
B=[6 7 6]
A\B
0 0 0
0 0 0
2.0000 2.3333 2.0000
If ve use MLS:
C = inv (A'*A) singular matrix
C = pinv(A'*A)
0.0051 0.0102 0.0153
0.0102 0.0204 0.0306
0.0153 0.0306 0.0459
D= C*A'*B
0.4286 0.5000 0.4286
0.8571 1.0000 0.8571
1.2857 1.5000 1.2857
So results A\B and inv (A'*A)*A'*B are different...
Answers:
My MATLAB (R2010b) says quite a lot about what A\B
does:
mldivide(A,B)
and the equivalent A\B
perform matrix left division (back slash). A
and B
must be matrices that have the same number of rows, unless A
is a scalar, in which case A\B
performs element-wise division — that is, A\B = A.\B
.
If A
is a square matrix, A\B
is roughly the same as inv(A)*B
, except it is computed in a different way. If A
is an n
-by-n
matrix and B
is a column vector with n
elements, or a matrix with several such columns, then X = A\B
is the solution to the equation AX = B
. A warning message is displayed if A
is badly scaled or nearly singular.
If A
is an m
-by-n
matrix with m ~= n
and B
is a column vector with m
components, or a matrix with several such columns, then X = A\B
is the solution in the least squares sense to the under- or overdetermined system of equations AX = B
. In other words, X
minimizes norm(A*X - B)
, the length of the vector AX - B
. The rank k
of A
is determined from the QR decomposition with column pivoting. The computed solution X
has at most k
nonzero elements per column. If k < n
, this is usually not the same solution as x = pinv(A)*B
, which returns a least squares solution.
mrdivide(B,A)
and the equivalent B/A
perform matrix right division (forward slash). B
and A
must have the same number of columns.
If A
is a square matrix, B/A
is roughly the same as B*inv(A)
. If A
is an n
-by-n
matrix and B
is a row vector with n
elements, or a matrix with several such rows, then X = B/A
is the solution to the equation XA = B
computed by Gaussian elimination with partial pivoting. A warning message is displayed if A
is badly scaled or nearly singular.
If B
is an m
-by-n
matrix with m ~= n
and A
is a column vector with m
components, or a matrix with several such columns, then X = B/A
is the solution in the least squares sense to the under- or overdetermined system of equations XA =