5.4 The PLU Factorization
As indicated above, some matrices don’t have an LU factorization. Here is an example.
In this case, there is another factorization which is useful called a PLU factorization. Here P is a
Example 5.4.1 Find a PLU factorization for the above matrix in 5.1.
Proceed as before trying to find the row echelon form of the matrix. First add −1
times the first row to the second row and then add −4 times the first to the third. This
There is no way to do only row operations involving replacing a row with itself added to a multiple
of another row to the second matrix in such a way as to obtain an upper triangular matrix.
Therefore, consider M with the bottom two rows switched.
Now try again with this matrix. First take −1 times the first row and add to the bottom row and
then take −4 times the first row and add to the second row. This yields
The second matrix is upper triangular and so the LU factorization of the matrix M′
Thus M′ = PM = LU where L and U are given above. Therefore, M = P2M = PLU and
This process can always be followed and so there always exists a PLU factorization of a given
matrix even though there isn’t always an LU factorization.
Example 5.4.2 Use a PLU factorization of M ≡
to solve the system
= b where b
Let Ux = y and consider PLy = b. In other words, solve,
Then multiplying both sides by P gives
Now Ux = y and so it only remains to solve