CHAPTER 09.07: ADEQUACY OF SOLUTIONS: Relating changes in coefficient matrix to changes in solution vector
In this segment we will talk about, how if you make certain changes in the coefficient matrix how they are related to changes in the solution vector and we are going to do this to the norms of the matrixes involved. We are not going to show the proof of the theorem but we are going to talk about what it means. So let suppose somebody gives you an equation like this. A times X is equal to C. What you do in order to be able to figure out whether this set of equations is well conditioned or ill conditioned? Which means that if I make a small change on the right hand side or in my coefficient matrix do I get smaller or large changes in my solution vector.
What we are going to do is in this case, we are going to change our coefficient matrix a little bit. And we will call the new matrix A prime and then we will get a different solution for that. Because if we are going to change the coefficient matrix and keep the other side the same then we are surely going to get a new solution vector. Now if we define our delta A matrix as the coefficient matrix has changed minus the original coefficient matrix. We define our delta X as the changed in the solution vector as the solution vector as we get here and the solution vector we get from the original set of equations. If we define that then there is a theorem which relates the norms of some of these matrixes. It is given as follows: that the lI? XIl divided by lI? X+? Il is less than equal to lI AIl times lI A^-1Il times ll? All over lIAIl so we are basically saying here is that the relative change in the solution vector and this is the relative change in the coefficient matrix in terms of the norm.
So the relative change in the norm of the solution vector is related to the relative change in the coefficient matrix by this quantity right here. So what that basically means is that if you make a relative change in your coefficient matrix the relative change in the solution vector can get amplified by the as much as this number right here. It doesn’t mean it gets amplified by that number but because there is a less than or equal to sign here that it can get amplified by as much as by this number here, which is simply the I AIl times lI A^-1Il so the amplification factor that you are seeing here is finding the of coefficient matrix, taking its inverse and then find in the of that multiplying the two and you get I AIl times lI A^-1Il. So this I AIl times lI A^-1Il is actually defined as the conditioned of the A matrix. Conditioned number of A. Which makes sense because when we are talking about well conditioning and ill conditioning system of equations and this is going to be the amplification factor for the relative change of the norm of the solution vector to the relative change in the coefficient matrix then if this is the amplification factor that is what is going to determine how much change that is going to be and that is why we call this quantity here to be the conditioned number of the matrix. And now how this is related to quantifying the relative change in the solution vector to the relative change in the coefficient matrix, we will talk about in a different segment. And that the end of this segment.