Matrix Equation

A matrix equation is an equation involving matrices, with one or more unknown matrices or vectors. A common example is the linear system

(1)

for a given matrix and vector . If is square and its determinant is nonzero, the unique solution is obtained using the matrix inverse,

(2)

This solution is nonzero iff . A singular matrix can also yield solutions, but they are not unique when they exist. For example, the equations and have every as a solution. More generally, a rectangular or square system is consistent iff the matrix rank of equals that of the augmented matrix . The solutions then consist of any one solution plus arbitrary elements of the null space of .

Numerical methods generally avoid forming the matrix inverse explicitly. They include Gaussian elimination and LU decomposition. For a symmetric positive definite matrix, the square root method can be used.

For a homogeneous matrix equation

(3)

to be solved for the s, consider the determinant

(4)

Now multiply by , which is equivalent to multiplying the first column (or any column) by ,

(5)

The value of the determinant is unchanged if multiples of columns are added to other columns. So add times column 2, ..., and times column to the first column to obtain

(6)

But from the original matrix, each of the entries in the first columns is zero since

(7)

so

(8)

Therefore, if there is an which is a solution, the determinant is zero. This is also true for , ..., , so a nonzero solution vector can exist only if the determinant is zero. Conversely, a square singular matrix has a nonzero null space, so its homogeneous system has a nonzero solution. This approach is the basis for Cramer's rule.

Given a numerical solution to a matrix equation, the solution can be iteratively improved using the following technique. Assume that the numerically obtained solution to

(9)

is , where is an error term. The first solution therefore gives

(10)

(11)

where is found by solving (10)

(12)

Combining (11) and (12) then gives

(13)