HP 15c - Manual (Page 82)

82
Section 4:
Using Matrix Operations
Matrix algebra is a powerful tool. It allows you to more easily formulate and solve many
complicated problems, simplifying otherwise intricate computations. In this section you will
find information about how the HP-15C performs certain matrix operations and about using
matrix operations in your applications.
Several results from numerical linear algebra theory are summarized in this section. This
material is not meant to be self-contained. You may want to consult a reference for more
complete presentations.
**
Understanding the LU Decomposition
The HP-15C can solve systems of linear equations, invert matrices, and calculate
determinants. In performing these calculations, the HP-15C transforms a square matrix into a
computationally convenient form called the LU decomposition of the matrix.
The LU decomposition procedure factors a square matrix A into the matrix product LU. L is
a lower-triangular matrix with 1's on its diagonal and with subdiagonal elements (those
below the diagonal) between -1 and +1, inclusive. U is an upper-triangular matrix.
††
For
example:
LU A






















5 . 0
3 2
1 5 .
0 1
1 1
3 2
.
Some matrices can't be factored into the LU form. For example,
LU A








2 1
1 0
for any pair of lower- and upper-triangular matrices L and U. However, if rows are
interchanged in the matrix to be factored, an LU decomposition can always be constructed.
Row interchanges in the matrix A can be represented by the matrix product PA for some
square matrix P. Allowing for row interchanges, the LU decomposition can be represented by
the equation PA = LU. So for the above example,
LU PA


































1 0
2 1
1 0
0 1
1 0
2 1
2 1
1 0
0 1
1 0
.
** Two such references are
Atkinson, Kendall E., An Introduction to Numerical Analysis, Wiley, 1978.
Kahan, W. "Numerical Linear Algebra," Canadian Mathematical Bulletin, Volume 9, 1966, pp. 756-801.
††
A lower-triangular matrix has 0’s for all elements above its diagonal. An uppertriangular matrix has 0's for all elements
below its diagonal.