description/proof of that for principal integral domain, rectangle matrix over domain, and invertible square matrix over domain, sum of principal ideals by specified-dimensional subdeterminants of product is sum of principal ideals by same dimensional subdeterminants of rectangle matrix
Topics
About: ring
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
Starting Context
- The reader knows a definition of principal integral domain.
- The reader admits the proposition that for any principal integral domain, any rectangle matrix over the domain, and any rectangular-matrix-columns-dimensional or rectangular-matrix-rows-dimensional square matrix over the domain, the sum of the principal ideals by the specified-dimensional subdeterminants of the product is contained in the sum of the principal ideals by the same-dimensional subdeterminants of the rectangle matrix.
Target Context
- The reader will have a description and a proof of the proposition that for any principal integral domain, any rectangle matrix over the domain, and any invertible rectangular-matrix-columns-dimensional or rectangular-matrix-rows-dimensional square matrix over the domain, the sum of the principal ideals by the specified-dimensional subdeterminants of the product is the sum of the principal ideals by the same-dimensional subdeterminants of the rectangle matrix.
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:
\(R\): \(\in \{\text{ the principal integral domains }\}\)
\(M\): \(\in \{\text{ the m x n matrices over } R\}\)
\(N\): \(\in \{\text{ the invertible n x n matrices over } R\}\)
\(O\): \(\in \{\text{ the invertible m x m matrices over } R\}\)
\(k\): \(\in \mathbb{N}\), \(1 \le k \le min (m, n)\)
\(I_k (M)\): \(= \text{ the sum of the principal ideals by the } k \text{ -dimensional subdeterminants of } M\)
\(I_k (M N)\): \(= \text{ the sum of the principal ideals by the } k \text{ -dimensional subdeterminants of } M N\)
\(I_k (O M)\): \(= \text{ the sum of the principal ideals by the } k \text{ -dimensional subdeterminants of } O M\)
//
Statements:
\(I_k (M N) = I_k (M)\)
\(\land\)
\(I_k (O M) = I_k (M)\)
//
As an immediate corollary, \(I_k (O M N) = I_k (M)\): \(I_k (O M N) = I_k (M N) = I_k (M)\).
2: Natural Language Description
For any principal integral domain, \(R\), any \(m x n\) matrix over \(R\), \(M\), any invertible \(n x n\) matrix over \(R\), \(N\), any invertible \(m x m\) matrix over \(R\), \(O\), any natural number, \(k\), such that \(1 \le k \le min (m, n)\), the sum of the principal ideals by the \(k\)-dimensional subdeterminants of \(M\), \(I_k (M)\), the sum of the principal ideals by the \(k\)-dimensional subdeterminants of \(M N\), \(I_k (M N)\), and the sum of the principal ideals by the \(k\)-dimensional subdeterminants of \(O M\), \(I_k (O M)\), \(I_k (M N) = I_k (M)\) and \(I_k (O M) = I_k (M)\).
3: Proof
Generally, for any matrix, \(A\), \(I_k (A)\) denotes the sum of the principal ideals by the \(k\) -dimensional subdeterminants of \(A\).
\(I_k (M N) \subseteq I_k (M)\), by the proposition that for any principal integral domain, any rectangle matrix over the domain, and any rectangular-matrix-columns-dimensional or rectangular-matrix-rows-dimensional square matrix over the domain, the sum of the principal ideals by the specified-dimensional subdeterminants of the product is contained in the sum of the principal ideals by the same-dimensional subdeterminants of the rectangle matrix.
There is an inverse, \(N^{-1}\).
\(I_k (M) = I_k (M N N^{-1}) \subseteq I_k (M N)\), by the proposition that for any principal integral domain, any rectangle matrix over the domain, and any rectangular-matrix-columns-dimensional or rectangular-matrix-rows-dimensional square matrix over the domain, the sum of the principal ideals by the specified-dimensional subdeterminants of the product is contained in the sum of the principal ideals by the same-dimensional subdeterminants of the rectangle matrix.
So, \(I_k (M N) = I_k (M)\).
\(I_k (O M) \subseteq I_k (M)\), by the proposition that for any principal integral domain, any rectangle matrix over the domain, and any rectangular-matrix-columns-dimensional or rectangular-matrix-rows-dimensional square matrix over the domain, the sum of the principal ideals by the specified-dimensional subdeterminants of the product is contained in the sum of the principal ideals by the same-dimensional subdeterminants of the rectangle matrix.
There is an inverse, \(O^{-1}\).
\(I_k (M) = I_k (O^{-1} O M) \subseteq I_k (O M)\), by the proposition that for any principal integral domain, any rectangle matrix over the domain, and any rectangular-matrix-columns-dimensional or rectangular-matrix-rows-dimensional square matrix over the domain, the sum of the principal ideals by the specified-dimensional subdeterminants of the product is contained in the sum of the principal ideals by the same-dimensional subdeterminants of the rectangle matrix.
So, \(I_k (O M) = I_k (M)\).
