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 Language Reference

## HERMITE Function

reduces a matrix to Hermite normal form

HERMITE( matrix)

where matrix is a numeric matrix or literal.

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

If the argument is square and nonsingular, the result will be the identity matrix. In general the result satisfies the following four conditions (Graybill 1969, p. 120):

• 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.
Consider the following example (Graybill 1969, p. 288):
```     a={3  6  9,
1  2  5,
2  4 10};
h=hermite(a);
```
These statements produce
```         H             3 rows      3 cols    (numeric)

1         2         0
0         0         0
0         0         1
```
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.

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