What is Gaussian elimination?
The goal with Gaussian elimination is multi-fold, it can be as simple as creating a Upper triangular matrix, to get a matrix in row echelon form, or to find an inverse matrix.
What is an elementary matrix?
An elementary matrix corresponds to a single row operation applied to the identity operation. They are nice because we can use them to track corresponding modifications to a matrix A. Furthermore, the inverse of an elementary matrix is just the identity matrix with -c in the (i, j) slot.
What is LU factorization?
LU factorization involves factoring a matrix A = LU s.t. L is a lower triangular matrix and U is an upper triangular matrix with the diagonals being pivots of A.
If A cannot be LU factorized, it is not regular.
Why do we care about LU factorization?
Once have the matrices L and U, we can use them to solve systems of linear equation. Like so:
solve for c via forward sub. Then, solve for x using backwards sub.
What is the uniqueness of inverse mean?
An inverse of a square matrix, if it exists, is unique. This is great, since if we have an inverse to a system of equation, then our solution will be unique. A singular matrix means the determinate is 0. An inverse of a matrix is an inverse of a matrix and vice-versa.
What is the inverse of a product?
If we have matrices A and B, then their product is invertible like so:
This property can change with any number of matrices, which going back to our LU factorization, we can use this with our chain of elementary matrices, for which we have established it is easy to find its inverse.
What is Gauss Jordan elimination?
We use the gauss elimination method to convert the matrix A into the identity matrix. Meanwhile, we augment the matrix with the identity matrix, tracking all row operations done to A on the matrix I. Starting with A | I, we end with I | A