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What is an underdetermined linear system?

What is an underdetermined linear system?

In mathematics, a system of linear equations or a system of polynomial equations is considered underdetermined if there are fewer equations than unknowns (in contrast to an overdetermined system, where there are more equations than unknowns). The terminology can be explained using the concept of constraint counting.

How do you know if a matrix is underdetermined?

If the number of rows in the matrix A, i.e. the number of equations, is less than the number of columns, i.e. the number of unknowns, then the system is underdetermined.

What is the difference between overdetermined and underdetermined?

The overdetermined case occurs when the system has been overconstrained — that is, when the equations outnumber the unknowns. In contrast, the underdetermined case occurs when the system has been underconstrained — that is, when the number of equations is fewer than the number of unknowns.

What is linear least square problem?

Mathematically, linear least squares is the problem of approximately solving an overdetermined system of linear equations A x = b, where b is not an element of the column space of the matrix A. The approach is called linear least squares since the assumed function is linear in the parameters to be estimated.

How to solve an underdetermined linear system using the least squares method?

I have an underdetermined linear system, with 3 equations and four unknows. I also know an initial guess for these 4 unknows. The article I am reading says: We can solve the system using the least squares method, starting form a guess.

Which is the correct solution to the least squares problem?

Note thatanysolution of the normal equations (3) is a correct solution to our least squares problem. Most likely,A0Ais nonsingular, so there is a unique solution. IfA0Ais singular, still any solution to (3) is a correct solution to our problem.

How to find the least squares solution for AAT?

The least squares solution can be determined using the Moore-Penrose pseudoinverse: x = AT(AAT)−1y where it is assumed that the inverse of AAT exists. Royi’s answer discusses the case when AAT is singular. In any case, you do not need an initial guess. The solution you’ll get is the solution with the smallest norm of all possible solutions.

Which is the solution with the smallest norm of all possible solutions?

The solution you’ll get is the solution with the smallest norm of all possible solutions. @Matt L. solution is correct under the assumption A is full rank. If it is otherwise, the solution using the SVD is always well defined which minimizes both the norm of the error and the norm of the solution.