Trigonalization

The Trigonalisierung is a term used in linear algebra , a branch of mathematics . It denotes a similarity mapping of a square matrix to an upper triangular matrix . This is not possible for every square matrix, which is why matrices that are similar to an upper triangular matrix are called trigonalizable matrices. Correspondingly, a vector space endomorphism is called a trigonalizable endomorphism if there is an upper triangular matrix among its representation matrices.

There is a connection between trigonalizable matrices and trigonalizable endomorphisms: The trigonalizable matrices are the representation matrices of the trigonalizable endomorphisms.

Criteria for trigonalization

The following statements are equivalent and thus determine whether a matrix can be trigonalized:

• the matrix can be trigonalized over the body . ${\displaystyle A}$ ${\displaystyle K}$
• the matrix is similar to an upper triangular matrix. That is, there is an upper triangular matrix and an invertible matrix with . ${\displaystyle A}$ ${\displaystyle D}$ ${\displaystyle P}$ ${\displaystyle D=P^{-1}AP}$
• the characteristic polynomial of the matrix breaks down into linear factors over the body . ${\displaystyle A}$ ${\displaystyle K}$
• the minimal polynomial of the matrix breaks down into linear factors over the body . ${\displaystyle A}$ ${\displaystyle K}$
• the matrix has a Jordan normal form above the body . ${\displaystyle A}$ ${\displaystyle K}$

In particular, every square matrix can thus be trigonalized, since here every non-constant polynomial breaks down into linear factors. ${\displaystyle \mathbb {C} }$

Calculation of the upper triangular matrix

In order to calculate the upper triangular matrix we are looking for, we first calculate the matrix with which the similarity mapping is carried out. The following applies: ${\displaystyle D}$ ${\displaystyle P}$

${\displaystyle D=P^{-1}AP}$

Furthermore, and have the same eigenvalues . ${\displaystyle A}$ ${\displaystyle D}$

Since the characteristic polynomial of divides into linear factors, there is an eigenvalue and an associated eigenvector . This eigenvector is now added to a basis of . Let the matrix be the base change matrix for changing the base from the base to the unit base. With it can be calculated and the shape ${\displaystyle A}$ ${\displaystyle \lambda _{1}}$ ${\displaystyle v_{1}}$ ${\displaystyle v_{1},v_{2},\dots ,v_{n}}$ ${\displaystyle K^{n}}$ ${\displaystyle T_{1}}$ ${\displaystyle v_{1},v_{2},\dots ,v_{n}}$ ${\displaystyle T_{1}^{-1}AT_{1}}$

${\displaystyle T_{1}^{-1}AT_{1}={\begin{pmatrix}\lambda _{1}&d_{1,2}&\cdots &d_{1,n}\\0&&&\\\vdots &&A_{1}&\\0&&&\end{pmatrix}}}$

The following applies to the characteristic polynomial of the matrix . It therefore also breaks down into linear factors and can thus be trigonalized again itself. This process can now be continued until one has calculated. The resulting matrix is ​​exactly the triangular matrix . The matrix is the product of the basic change matrices. ${\displaystyle (n-1)\times (n-1)}$ ${\displaystyle A_{1}}$ ${\displaystyle p_{A}(\lambda )=(\lambda -\lambda _{1})p_{A_{1}}}$ ${\displaystyle A_{1}}$ ${\displaystyle A_{n-1}=d_{n,n}}$ ${\displaystyle D}$ ${\displaystyle P}$ ${\displaystyle T_{1}T_{2}\dots T_{n-1}}$

See also

• Schur decomposition is an example of a trigonalization procedure over or ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$
• Diagonalizable matrix

literature

• Gerd Fischer : Linear Algebra. An introduction for first-year students. 14th edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03217-0 .