Hence there exists a unique solution to the set of equations given by (1). If A is of full rank, that is, rank(A) = n, then the dimension of null space = n - n = 0. This is also known as overdetermined set of equations. More number of equations and less number of variables. You have many solutions even when rank(A) = r. If A is of full rank, that is, rank(A) = m, there exists a null space of dimension (n-m) and hence there would be many solutions to equation (1). This is also known as under-determined set of equations. L ess number of equations and more number of variables. Lets assume that we have a solution for the set of equations given by (1). Whenever \gamma > 0, or we have a non-trivial null space, we will have multiple solutions. Dimension of null space is given by \gamma = n - rank(A). Since xh is a vector that lies in the null space of A, we have A * xh = 0. Multiplying both sides of equation (2) by A, we getĪx = A * xp + \lambda A * xh = A * xp = y Xp is the particular solution and xh is the homogenous solution. General solution to equation (1) is given by
In case this rank condition is satisfied, we know that there exists at least one solution.
Solution to equation (1) exists if and only if y lies in the range space of A. The task is to find a solution ' x' that satisfies equation (1). ' x' is a n-dimensional vector and y is a m-dimensional output or measurement vector. Where A is m x n dimensional constant matrix. I would discuss this issue briefly so that I can refer back to it when ever needed.
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