1 LU decomposition
Suppose we have the system of equations
The motivation for an decomposition is based on the observation that systems of equations involving triangular coefficient matrices are easier to deal with. Indeed, the whole point of Gaussian elimination is to replace the coefficient matrix with one that is triangular. The decomposition is another approach designed to exploit triangular systems.
We suppose that we can write
where is a lower triangular matrix and is an upper triangular matrix. Our aim is to find and and once we have done so we have found an decomposition of .
Key Point 5
An decomposition of a matrix is the product of a lower triangular matrix and an upper triangular matrix that is equal to .
It turns out that we need only consider lower triangular matrices that have 1s down the diagonal. Here is an example. Let
where and .
Multiplying out and setting the answer equal to gives
Now we use this to find the entries in and . Fortunately this is not nearly as hard as it might at first seem. We begin by running along the top row to see that
Now consider the second row
Notice how, at each step, the equation being considered has only one unknown in it, and other quantities that we have already found. This pattern continues on the last row
We have shown that
and this is an decomposition of .
Task!
Find an decomposition of .
Let
then, comparing the left and right hand sides row by row implies that , , which implies and which implies that . Hence
is an decomposition of .
Task!
Find an decomposition of .
Using material from the worked example in the notes we set
and comparing elements row by row we see that
and it follows that
is an decomposition of the given matrix.