Trigonalizable mapping/Eigentheorie/No proof/Section

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Let denote a field, and let denote a finite-dimensional vector space. A linear mapping is called trigonalizable, if there exists a basis such that the describing matrix of with respect to this basis is an

upper triangular matrix.

Diagonalizable linear mappings are in particular trigonalizable. The reverse statement is not true, as example shows.

Let denote a field, and let denote an finite-dimensional vector space. Let

denote a

linear mapping. Then the following statements are equivalent.
  1. is trigonalizable.
  2. The characteristic polynomial has a factorization into linear factors.
If is trigonalizable and is described by the matrix with respect to some basis, then there exists an invertible matrix

such that is an upper triangular matrix.


This proof was not presented in the lecture.

Let denote a square matrix with complex entries. Then is trigonalizable.

This follows from fact and the Fundamental theorem of algebra.