Matlab, operator A/B

The manual is written:

A\B returns a least-squares solution to the system of equations A*x= B.

It means: x = inv (A'*A)*A'*B? However, the matrix A'*A is singular...

Let us suppose to have two row vectors:

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]

And

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... Could anybody wrote me a sequence of Matlab commands in the backgeound of A/B operation in this case?

[Matt J]

I think you really meant so say that pinv(A'*A)*A'*B and A\B are different. When A is singular inv (A'*A)*A'*B is undefined.

Answer 1

[Matt J]

When A is singular there are infinite least squares solutions to A*X=B and A\B will pick just one of them, while pinv(A)*B picks another.

A/B is equivalent to (B'\A')'