Triangular matrix#Triangularisability

The Trigonalisierung is a concept from linear algebra, a branch of mathematics. It denotes a similarity figure of a square matrix to an upper triangular matrix. This is not possible for each square matrix and, therefore, is referred to matrices similar to an upper triangular matrix, as trigonalisierbare matrices. Correspondingly, one refers to a vector space endomorphism as trigonalisierbaren endomorphism if there is an upper triangular matrix with its representation matrices.

Between trigonalisierbaren matrices and trigonalisierbaren endomorphisms there is a correlation: the trigonalisierbaren matrices are the representation matrices of trigonalisierbaren endomorphisms.

Criteria for Trigonalisierbarkeit

The following statements are equivalent and thus define whether a matrix is trigonalisierbar:

• The matrix is ​​trigonalisierbar over the body.
• The matrix is similar to an upper triangular matrix. That is, there is an upper triangular matrix, and an invertible matrix.
• The characteristic polynomial of the matrix breaks down over the body into linear factors.
• The minimal polynomial of the matrix breaks down over the body into linear factors.
• The matrix has on the body a Jordan normal form.

In particular, it is any square matrix over trigonalisierbar, since every polynomial decomposes into linear factors.

Calculation of the upper triangular matrix

To calculate the desired upper triangular matrix, we first calculate the matrix of the similarity mapping is performed. The following applies:

Furthermore, and have the same eigenvalues.

Since the characteristic polynomial of splits into linear factors, there is an eigenvalue and an associated eigenvector. This eigenvector is then added to a base of the. The matrix is according to the change of basis matrix for the change of basis from the standard basis. This can be calculated and the shape

For the characteristic polynomial of the matrix. It decomposes therefore into linear factors and is thus self again trigonalisierbar. This process goes on now until it has been calculated. The resulting matrix is exactly the triangle matrix. The matrix is the product of Basiswechselmatrizen.