Hermite and Smith forms

http://www.math.ubc.ca/~cass/siegel/Smith.pdf

https://en.wikipedia.org/wiki/Hermite_normal_form

SAS

HERMITE Function

HERMITE (matrix) ;

The HERMITE function uses elementary row operations to compute the Hermite normal form of a matrix. For square matrices this normal form is upper triangular and idempotent.

If the argument is square and nonsingular, the result is the identity matrix. In general the result satisfies the following four conditions (Graybill, 1969):

• It is upper triangular.
• It has only values of 0 and 1 on the diagonal.
• If a row has a 0 on the diagonal, then every element in that row is 0.
• If a row has a 1 on the diagonal, then every off-diagonal element is 0 in the column in which the 1 appears.

The following statements compute an example from Graybill (1969):

a = {3  6  9,
     1  2  5,
     2  4 10};
h = hermite(a);
print h;

http://support.sas.com/documentation/cdl/en/imlug/65547/HTML/default/viewer.htm#imlug_langref_sect260.htm

If the argument is a square matrix, then the Hermite normal form can be transformed into the row-echelon form by rearranging rows in which all values are 0.

The trace of the Hermite matirx is the rank.

Row vector is a function

A row vector is a function takes a column vector  as input and produce a scalar

LOESS/LOWESS Algorithm

https://www.weisang.com/en/documentation/loessandlowessalgorithm_en/

Lagrange Multipliers

We can connect Lagrange to Ridge and Lasso Regression.

http://www.math.harvard.edu/archive/21a_spring_09/PDF/11-08-Lagrange-Multipliers.pdf

Matrix Calculus from http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html

Re-Dock the Explorer/Results Window in SAS

Solution:

For Explorer window: Use menu View >> Contents Only.

For Results window: Use menu View >> Results.

multiple ampersand for SAS macro

When macro variables contain the name of another macro variable or pieces of names of other macro variables the resulting code will often contain two (or more!!) ampersands.