Scripting Guide - SAS

Chapter 7 Data Structures 161
Matrices
Another use of Cholesky() is in reordering matrices within Trace() expressions. Expressions such as
Trace(A*B*A`) might involve huge operations counts if A has many rows. However, if B is small and can
be factored into LL′ by Cholesky, then it can be reformulated to Trace(A*L*L`*A`). The resulting
matrix is equal to Trace(L`A`*AL). This expression involves a much smaller number of operations,
because it consists of only the sum of squares of the AL elements.
Use the function Chol Update() to update a Cholesky decomposition. If L is the Cholesky root of a nxn
matrix A, then after using cholUpdate(L, C, V), L will be replaced with the Cholesky root of
A+V*C*V'. C is an mxm symmetric matrix and V is an nxm matrix.
Examples
Manually update the Cholesky decomposition, as follows:
exS = [16 1 0 11 -1 12, 1 11 -1 1 -1 1, 0 -1 12 -1 1 0, 11 1 -1 11 -1 9, -1 -1
1 -1 9 -1, 12 1 0 9 -1 12];
// conducts the Cholesky decomposition
exAchol = Cholesky( exS);
// adds two column vectors to the design matrix
exV = [1 1, 0 0, 0 1, 0 0, 0 0, 0 1];
/* The first column vector is added to one of the rows in the design matrix.
The second column vector is subtracted from one of the rows in the design
matrix. */
exC = [1 0, 0 -1];
// updates the Cholesky decomposition manually
exAnew = exS + exV * exC * exV`;
exAcholnew = Cholesky( exAnew);
Instead of manual updating, use Chol Update() to update the Cholesky decomposition, as follows:
// updates the Cholesky decomposition more efficiently
exAcholnew_test =
Chol Update( exAchol, exV, exC );
// results are the same as the manual process
Show( exAcholnew_test );
Show( exAcholnew );
Singular Value Decomposition
The SVD() function finds the singular value decomposition of a matrix. That is, for a matrix A, SVD()
returns a list of three matrices U, M, and V, so that U*diag(M)*V`=A.
Note the following:
• When A is taller than it is wide, M is more compact, without extra zero diagonals.