backslash (\)

Scilab Function
Last update : 25/04/2007

backslash (\) - left matrix division.

### Calling Sequence

  `x=A\b`

### Description

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 /.

### Examples

    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

### See Also

`slash` , `inv` , `pinv` , `percent` , `ieee` ,