This canonical form of matrix called

Question

I am reading Rao, R.A., & Bhimasankaram, P. (2000). Linear algebra, and it has a habit of using non-standard terminology. This is what it calls Hermite canonical form:

A square matrix $\mathbf H$ of order $n$ is said to be in Hermite
canonical form iff:

1. $\mathbf H$ is upper triangular,
2. each diagonal entry of $\mathbf H$ is $0$ or $1$,
3. the $i$-th row of $\mathbf H$ is null if $\mathbf H(i,i) = 0$, and
4. the $i$-th column of $\mathbf H$ is $\mathbf e_i$ if $\mathbf H(i,i) = 1$.

This seems to be the same as the definition of reduced echelon form with the exception of the constraint that the matrix is square and (3.). However, when I look up anything related to Hermite canonical form, I get something to do with matrix over integers, and a completely different definition. So either the authors are using that definition from a different perspective, or Hermite canonical form refers to two different things, or they are calling something completely different Hermite canonical form.

The authors use this matrix in checking consistency of a system, by augmenting the output vector to the coefficient matrix, and performing the algorithm to obtain its "Hermite" form. If the "Hermite" form of the coefficient matrix has a null row, but the corresponding entry in the transformed output column is not $0$, then the system is inconsistent; and if it is consistent, a general solution of the system is:
$$
\mathbf d + (\mathbf I – \mathbf H) \mathbf z
$$
where $\mathbf d$ is the transformed form of the output column (and in fact a particular solution), $\mathbf H$ is the "Hermite" form of the coefficient matrix if it is square, $\mathbf z$ is any suitable vector. If the matrix is non square, they apply some different algorithm where they get a similar (as in the English similar) non-square matrix and exclude extra rows (if any), but that's too involved, so I will ignore that.

(edit)
If the matrix is not square, it seems they just act as if there were some extra null columns/row to make it square.

Also if we augment an identity matrix, the transformed identity matrix will correspond to the generalised inverse of the coefficient matrix.

Answer 1

It seems the terminology may not be totally standard.

Glancing through my bookshelf, in Nering's Linear Algebra and Matrix Theory, 2nd ed. (1970), we have on p. 54

Theorem 5.1. Given any $m\times n$ matrix $A$ of rank $\rho$, there exists a non-singular $m\times m$ matrix $Q$ such that $A'=Q^{-1}A$ has the following form:
(1) There is at least one non-zero element in each of the first $\rho$ rows of $A'$, and the elements in all remaining rows are zero.
(2) The first non-zero element appearing in row $i$ ($i\le\rho$) is a $1$ appearing in column $k_i$ where $k_1<k_2<\cdots<k_{\rho}$.
(3) In column $k_i$ the only non-zero element is the $1$ in row $i$.

The author then writes

We say that a matrix satisfying the conditions of Theorem 5.1 is in Hermite normal form. Often this form is called a row-echelon form. And sometimes the term, Hermite normal form, is reserved for a square matrix containing exactly the numbers that appear on the form we obtained in Theorem 5.1 with the change that row $i$ beginning with a $1$ in column $k_i$ is moved down to row $k_i$. Thus each non-zero row begins on the main diagonal and each column with a $1$ on the main diagonal is otherwise zero.

The latter form seems to align with what your book calls Hermite canonical form.