Triangular matrix

Under a triangular matrix is understood in mathematics, a square matrix, which is characterized in that all entries below (upper triangular matrix ) and above ( lower triangular matrix ) the main diagonal are zero. If, in addition, the entries on the main diagonal are all zero, then one speaks of a real or strict triangular matrix.

Triangular matrices play an important role, which is based on dividing up a matrix into the product of an upper and a lower triangular matrix, among others, in solving linear systems of equations using the LU decomposition.

• 5.1 Algebraic properties

Examples

Upper and lower triangular matrix

A matrix is ​​called upper triangular if all entries below the main diagonal are. For the entries on the main diagonal itself, there are no restrictions.

Thus applies for an upper triangular matrix:

The analogy is a matrix lower triangular if all entries above the main diagonal are

Trigonalisierbarkeit

Main article: Trigonalisierung.

If V is a vector space over the field and you have a square matrix, which is the representation of a linear map ( vector space endomorphism ), is the name given trigonalisierbar if she has seen in a different basis, an upper triangular, so is trigonal. So Wanted is a trigonal matrix that is too similar.

This is the case, if the characteristic polynomial is divided to the body in the linear factors.

If, as each matrix is trigonalisierbar, because after the Fundamental Theorem of Algebra, the body is algebraically closed.

Strict upper and lower triangular matrix

There are two different definitions for the term strictly upper triangular matrix, depending on whether one considers general or only invertible matrices. The former are nilpotent, unipotent latter. The following definitions are analogous for strict lower triangular matrices.

Nilpotent triangular matrices

In a strict upper triangular matrix in this sense, all entries are either below or on the main diagonal of the matrix. It is thus:

In a matrix that is true.

Unipotent triangular matrices

In a strict upper triangular matrix in the sense of invertible matrices are all entries below the main diagonal of the matrix, while the diagonal entries are all equal to 1. It is thus:

A thus looks like this: .

Such a matrix is the special case of a unipotent matrix, ie the matrix is ​​nilpotent, so there is a number such that:

Properties

It can be proved:

• The product of lower (upper) triangular matrices is again lower ( upper) triangular matrix.
• The product of strict lower (upper) triangular matrices is again a strict lower (upper) triangular matrix.
• The inverse of an invertible lower (upper) triangular matrix is a lower (upper) triangular matrix.
• The determinant of a triangular matrix is the product of its main diagonal elements.
• The eigenvalues ​​of a triangular matrix, the elements of the main diagonal.

Algebraic properties

• The set of all upper triangular matrices forms a solvable Lie algebra, the set of all nilpotent upper triangular matrices a nilpotent Lie algebra.
• The set of all invertible upper triangular matrices forms a solvable group, the set of all unipotent upper triangular matrices a nilpotent group.
• The number of elements of a triangular matrix, which may be different from zero; this is the dimension of the Lie group and algebra group.

Use of triangular matrices

Because of their special properties are triangular matrices at different locations, in particular also in the process of numerical mathematics. In the following table the body ( algebra) is assumed.

• In the Jordan normal form of a matrix is ​​similarly transformed to a triangular shape that is almost diagonal.
• In the Schur normal form, a matrix is ​​unitarily similar transformed into a triangular matrix. The method QR calculated it numerically.
• For a regular matrix of the Gaussian algorithm calculates an LU decomposition into the product of a lower ( left ) and an upper triangular matrix ( right ).
• The Gram -Schmidt orthogonalization with column vectors of a matrix calculates the QR decomposition of the matrix into the product of an orthogonal matrix and an upper ( right ) triangular matrix.