Linear Algebra part 2 - Inverses and LU
2008-04-02
Matrix Inverses
As a quick review of the method of matrix multiplication, let's go over what matrices can be multiplied. If you have a matrix of size m x n, where m is the columns and n is the rows, you can multiply this by a matrix of size n x p. The resulting matrix will be of size m x p. The basic equation of matrix multiplication is
(AB)i,j = Σr=1 to nAi,nBn,j
Now to inverses..Not all matrices have inverses. If there is an inverse than,
A-1A = I = AA-1
Where I is equal to the identity matrix. If the matrix does not have an inverse than it is a singular matrix. If this is the case, than a vector other than the zero vector, will take the matrix Ax = 0. This means A is not inveritable. If there does exist an inverse, we can find it through Gauss-Jordan elimination. This is a similar process as the Gauss elimination. Let's go through this with a 2x2 matrix. First, we augment this by the identity matrix...
Now we do forward elimination to get the left side of the matrix to upper triangular form. So we take 2 * row1 and subtract from row2.
Now use elimination upward to get rid of that 3 in the 1,2 position. Multiply 3*row2 and subtract that from row1.
Now we see that we are left with I | A
-1.
LU Factorization
LU factorization is taking a matrix A into its Lower and Upper triangular forms. This is closely related to the Elimination matrices we saw before. In fact, we will see that the U is simply the Gauss elimination and the L is the inverse of the E's. Given the matrix,
We can do E
2,1A = U where E and U are...
Now we have to take the inverse of E
2,1 and bring it over to the other side to produce the L. Because the inverse of the E matrix is simply the negative of the 2,1 element, it is easy to see that the A = LU factorization is...
Generally speaking for a 3x3 matrix you can use the formula,
A = E
2,1-1E
3,1-1E
3,2-1U... = LU.