Scripting Guide - SAS

160 Data Structures Chapter 7
Matrices
newA=evecs*diag(evals)*evecs`;
Note the following about eigenvalues and eigenvectors:
• The eigenvector matrices are orthonormal, such that the inverse is the transpose: E′E = EE′ = I .
• Eigenvectors are uniquely determined only if the eigenvalues are unique.
• Zero eigenvalues correspond to singular matrices.
• Inverses can be obtained by inverting the eigenvalues and reconstituting with the eigenvectors.
Moore-Penrose generalized inverses can be formed by inverting the nonzero eigenvalues and
reconstituting. (See “GInverse” on page 156.)
Note: You must decide whether a very small eigenvalue is effectively zero.
• The eigenvalue decomposition enables you to view any square-matrix multiplication as the following:
– a rotation (multiplication by an orthonormal matrix)
– a scaling (multiplication by a diagonal matrix)
– a reverse rotation (multiplication by the orthonormal inverse, which is the transpose), or in notation:
A*x = E`*diag(M)*E*x; // E rotates, diag(M) scales, E` reverse-rotates
Cholesky Decomposition
The Cholesky() function performs Cholesky decomposition. A positive semi-definite matrix A is
re-expressed as the product of a nonsingular, lower-triangular matrix L and its transpose: L*L' = A.
E=[ 5 4 1 1, 4 5 1 1, 1 1 4 2, 1 1 2 4];
L=Cholesky(E);
[2.23606797749979 0 0 0,
1.788854381999832 1.341640786499874 0 0,
0.447213595499958 0.149071198499986 1.9436506316151 0,
0.447213595499958 0.149071198499986 0.914659120760047 1.71498585142509]
To verify the results, enter the following:
L*L`;
[5 4 1 1,
4 5 1 1,
1 1 4 2,
1 1 2 4]
About Cholesky Decomposition
Cholesky() is useful for reconstituting expressions into a more manageable form. For example, eigenvalues
are available only for symmetric matrices in JMP, so the eigenvalues of the product AB could not be
obtained directly. (Although A and B can be symmetric, the product is not symmetric.) However, you can
usually rephrase the problem in terms of eigenvalues of L′BL where L is the Cholesky root of A, which has
the same eigenvalues.