Howtosignalprocessing
- What is the minimum norm solution?
- How many solutions does an underdetermined system have?
- How do you find the solution of an underdetermined system?
What is the minimum norm solution?
A vector x∗ satisfying Ax∗ = b is the minimum-norm solution to the system of equations Ax = b if and only if x∗ · y = 0 for all solutions y of the homogeneous system Ay = 0.
How many solutions does an underdetermined system have?
In general, an underdetermined system of linear equations has an infinite number of solutions, if any. However, in optimization problems that are subject to linear equality constraints, only one of the solutions is relevant, namely the one giving the highest or lowest value of an objective function.
How do you find the solution of an underdetermined system?
This underdetermined system has infinitely many solutions, but we are seeking a solution x∗ such that x∗ is minimized. To obtain this solution, we need to characterize the non-uniqueness of solutions of Ax = b. x = x0 − y, where x0 is a particular solution of Ax = b, and y ∈ N(A); that is, Ay = 0. z · Ay = z · 0 = 0.
