definition of rank of matrix over ring
Topics
About: matrices space
The table of contents of this article
Starting Context
- The reader knows a definition of determinant of square matrix over ring.
Target Context
- The reader will have a definition of rank of matrix over ring.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( R\): \(\in \{\text{ the rings }\}\)
\( m\): \(\in \mathbb{N} \setminus \{0\}\)
\( n\): \(\in \mathbb{N} \setminus \{0\}\)
\( M'\): \(\in \{\text{ the } m \times n R \text{ matrices }\}\)
\(*k\): \(\in \mathbb{N} \setminus \{0\}\)
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Conditions:
\(k = max \{k' \in \{1, ..., min (\{m, n\})\} \vert \exists M \in \{\text{ the } k \times k \text{ submatrices of } M'\} (det M \neq 0)\}\)
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