definition of eigenvalues of square 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 eigenvalues of square 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:
\( R\): \(\in \{\text{ the rings }\}\)
\( n\): \(\in \mathbb{N} \setminus \{0\}\)
\( M\): \(\in \{\text{ the } n \times n R \text{ matrices }\}\)
\(*S\): \(= \text{ the set of the roots for } \lambda \text{ of } det (M - \lambda I) = 0\)
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Conditions:
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2: Note
When \(R\) is any field, \(S\) has at most \(n\) elements including any duplicate roots, by the proposition that over any field, any n-degree polynomial has at most n roots: \(det (M - \lambda I)\) is an \(n\)-degree polynomial.
When \(R = \mathbb{C}\), \(S\) has exactly \(n\) elements including any duplicate roots, as a well-known fact.
When \(R = \mathbb{C}\) and \(M\) is a Hermitian matrix, \(S \subseteq \mathbb{R}\), as a well-known fact.
When \(R = \mathbb{R}\) and \(M\) is a symmetric matrix, \(S\) has exactly \(n\) elements including any duplicate roots: regarding \(R = \mathbb{C}\), \(S\) has exactly \(n\) complex elements including any duplicate elements, as is mentioned above, but \(M\) is Hermitian, so, the elements are all real, so, \(S\) has exactly \(n\) real elements including any duplicate elements.
