Mathematics - The MathWorks - #25

general system of simultaneous equations. The two division symbols, slash, /, and backslash, \, are used for the two situations where the unknown matrix appears on the left or right of the coefficient matrix:

X = B/A X = A\B

Denotes the solution to the matrix equation XA = B.

Denotes the solution to the matrix equation AX

= B.

You can think of "dividing" both sides of the equation AX = B or XA = B by A. The coefficient matrix A is always in the "denominator."

The dimension compatibility conditions for X = A\B require the two matrices A and B to have the same number of rows. The solution X then has the same number of columns as B and its row dimension is equal to the column dimension of A.For X = B/A, the roles of rows and columns are interchanged.

In practice, linear equations of the form AX = B occur more frequently than those of the form XA = B. Consequently, the backslash is used far more frequently than the slash. The remainder of this section concentrates on the backslash operator; the corresponding properties of the slash operator can be inferred from the identity:

(B/A)' = (A'\B')

The coefficient matrix A need not be square. If A is m-by-n, there are three cases:

m = n Square system. Seek an exact solution.

m>n Overdetermined system. Find a least squares

solution.

m<n Underdetermined system. Find a basic solution

with at most m nonzero components.

The backslash operator employs different algorithms to handle different kinds of coefficient matrices. The various cases, which are diagnosed automatically by examining the coefficient matrix, include: