backslash - (\) left matrix division.
Backslash denotes left matrix division. x=A\b is a solution to A*x=b .
If A is square and nonsingular x=A\b (uniquely defined) is equivalent to x=inv(A)*b (but the computations are much cheaper).
If A is not square, x is a least square solution. i.e. norm(A*x-b) is minimal (euclidian norm). If A is full column rank, the least square solution, x=A\b , is uniquely defined (there is a unique x which minimizes norm(A*x-b) ). If A is not full column rank, then the least square solution is not unique, and x=A\b , in general, is not the solution with minimum norm (the minimum norm solution is x=pinv(A)*b ).
A.\B is the matrix with (i,j) entry A(i,j)\B(i,j) . If A (or B ) is a scalar A.\B is equivalent to A*ones(B).\B (or A.\(B*ones(A))
A\.B is an operator with no predefined meaning. It may be used to define a new operator (see overloading) with the same precedence as * or /.
A=rand(3,2);b=[1;1;1]; x=A\b; y=pinv(A)*b; x-y A=rand(2,3);b=[1;1]; x=A\b; y=pinv(A)*b; x-y, A*x-b, A*y-b A=rand(3,1)*rand(1,2); b=[1;1;1]; x=A\b; y=pinv(A)*b; A*x-b, A*y-b A=rand(2,1)*rand(1,3); b=[1;1]; x=A\b; y=pinv(A)*b; A*x-b, A*y-b