LLS solves linearly constrained least squares (or LCLS) problems, which have the form:
\begin{array}{ll}
\mbox{minimize} & \|Ax - b\|_2^2 \\
\mbox{subject to} & Cx = d
\end{array}
where the unknown variable x is a vector of size n. The values for A, b, C, and d are given and have sizes m\times n, m, p\times n, and p, respectively. LLS finds a value for x that satisfies the linear equality constraints Cx = d and minimizes the objective, the sum of the squares of the entries of Ax - b.
When there are no equality constraints, LCLS reduces to the simple unconstrained least squares problem (LS):
\begin{array}{ll}
\mbox{minimize} & \|Ax-b\|_2^2
\end{array}.
When the objective is absent, LCLS reduces to finding x that satisfies Cx=d, i.e., solving a set of linear equations.
There is a unique solution to the LCLS problem if and only if there is a unique solution to the following system of linear equations in the variable x and a new variable z:
\begin{bmatrix} 2A^TA & C^T \\ C & 0 \end{bmatrix}
\begin{bmatrix} x \\ z \end{bmatrix} =
\begin{bmatrix} 2A^Tb \\ d \end{bmatrix},
i.e., the matrix on the left is invertible. This occurs when the matrix C has independent rows, and the matrix \begin{bmatrix} A\\ C\end{bmatrix} has indepedent columns.
When there are no equality constraints, the unconstrained least squares problem has a unique solution if and only if the system of linear equations:
2A^TA x = 2A^Tb
has a unique solution, which occurs when A^TA is invertible, i.e., the columns of A are independent.
When the objective is absent, the system of linear equations Cx = d has a unique solution if and only if C is invertible.
LLS allows you to specify an LCLS problem in a natural way. It translates your specification into the general form in this section, and then solves the appropriate set of linear equations.