On Wednesday, last week, we completed our material over elementary matrices, using them to derive the inverse of a matrix. Upon proving that a matrix is non-singular (i.e., invertible) if and only if it is row equivalent to the identity, we noticed that the same row operations that change a matrix A into the identity will change the identity into the inverse of A.
We, then, closed off that section by looking at how to form an LU decomposition, or factorization, of a matrix. The basic algorithm we used to row reduce a matrix to upper triangular form keeping track of the elementary matrices used, then computing L as the product of the inverses of those elementary matrices. Unfortunately, I forgot to mention (and WILL mention in the next class) that this only works if we use only the row operation of type III. That fact guarantees that the product of the inverses of those elementary matrices will be lower triangular.
After finishing section 4, we started to talk about partitioning matrices. We are simply trying to show that all of our matrix algebra works the same when the elements of the matrices are also matrices. I’m having a hard time convincing the students of the significance of this, since in any given matrix calculation, it is just as easy to compute the sums and products with simple matrices as with block matrices. As I see it, the greatest benefit to using block matrices is to deal with matrices that have a specific block structure that is maintained after calculation.
Next time, we will finish this section and be ready to schedule an exam.
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