Triangular matrix

In mathematics, a triangular matrix is understood to be a square matrix which is characterized by the fact that all entries below (upper triangular matrix) or above (lower triangular matrix) the main diagonal are zero. If the entries on the main diagonal are all zero, one speaks of a real or strict triangular matrix .

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

Examples

The following matrices are examples of triangular matrices:

${\displaystyle {\begin{pmatrix}a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&0&a_{33}\end{pmatrix}}\quad {\text{mit}}\quad a_{ij}\in \mathbb {R} }$

${\displaystyle {\begin{pmatrix}3&2&3&4\\0&5&5&6\\0&0&0&7\\0&0&0&9\end{pmatrix}}}$.

Upper and lower triangular matrix

A matrix is ​​called an upper triangular matrix if all entries below the main diagonal are zero. There are no restrictions on the entries on the main diagonal itself.

For an upper triangular matrix the following applies: ${\displaystyle A=(a_{ij})}$

${\displaystyle i>j\,\Rightarrow \,a_{ij}=0}$.

Similarly, a matrix is ​​called a lower triangular matrix if all entries above the main diagonal are equal to zero, i.e. if applies

${\displaystyle i<j\,\Rightarrow \,a_{ij}=0}$.

Normalization

A triangular matrix is called a normalized triangular matrix if all diagonal entries are equal to 1: ${\displaystyle A=(a_{ij})}$

${\displaystyle a_{ii}=1}$for everyone . ${\displaystyle i}$

Trigonalizability

→ Main article : Trigonalization

If a vector space is above the body and a square matrix is ​​available , which is the representation of a linear mapping (vector space endomorphism ), then this is said to be trigonalizable if it has an upper triangular shape when viewed in a different basis . We are looking for a triangular matrix that is similar to . ${\displaystyle V}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle A}$ ${\displaystyle f\colon V\to V}$ ${\displaystyle B}$ ${\displaystyle A}$

This is precisely the case if the characteristic polynomial of over the body breaks down into linear factors . ${\displaystyle A}$ ${\displaystyle \mathbb {K} }$

If , every matrix can be trigonalized, since according to the fundamental theorem of algebra the field is algebraically closed. ${\displaystyle \mathbb {K} =\mathbb {C} }$ ${\displaystyle \mathbb {C} }$

Strict upper and lower triangular matrix

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

Nilpotent triangular matrices

With a strict upper triangular matrix in this sense, all entries are both below and on the main diagonal of the matrix . The following applies: ${\displaystyle 0}$

${\displaystyle i\geq j\,\Rightarrow \,a_{ij}=0.}$

For a matrix, the following applies . ${\displaystyle n\times n}$ ${\displaystyle A^{n}=0}$

Unipotent triangular matrices

With a strict upper triangular matrix in the sense of invertible matrices, all entries are below the main diagonal of the matrix , while the diagonal entries are all the same (cf. normalized triangular matrix above). The following applies: ${\displaystyle 0}$ ${\displaystyle 1}$

${\displaystyle i>j\,\Rightarrow \,a_{ij}=0}$

${\displaystyle i=j\,\Rightarrow \,a_{ij}=1}$

Such a matrix is thus as follows: . ${\displaystyle n\times n}$ ${\displaystyle A}$ ${\displaystyle A={\begin{pmatrix}1&b_{1,2}&\cdots &b_{1,n}\\0&\ddots &\ddots &\vdots \\\vdots &\ddots &\ddots &b_{n-1,n}\\0&\cdots &0&1\end{pmatrix}}}$

Such a matrix is the special case of a unipotent matrix ; H. the matrix is nilpotent , so there is a number such that: ${\displaystyle A}$ ${\displaystyle A-I}$ ${\displaystyle m}$

${\displaystyle (A-I)^{m}=0}$.

properties

It can be proven:

• The product of lower (upper) triangular matrices is again a 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 are 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 that can be different from zero is ; this is also the dimension as a Lie group or an algebraic group . ${\displaystyle n\times n}$ ${\displaystyle {\tfrac {n(n+1)}{2}}}$

Use of triangular matrices

Because of their special properties, triangular matrices are used in various places, especially in numerical mathematics . The following list is based on the body . ${\displaystyle \mathbb {C} }$

• In the case of a regular matrix , the Gaussian algorithm calculates an LR decomposition into the product of a normalized lower (left) triangular matrix and an upper (right) triangle matrix for a suitable permutation matrix . ${\displaystyle A}$ ${\displaystyle P}$ ${\displaystyle PA=LR}$ ${\displaystyle L}$ ${\displaystyle R}$
• The QR decomposition of a matrix into a unitary matrix and an upper triangular matrix can be calculated using household transformations , Givens rotations or the Gram-Schmidt orthogonalization method, among other things . ${\displaystyle A=QR}$ ${\displaystyle A}$ ${\displaystyle Q}$ ${\displaystyle R}$
• In the Jordan normal form , a matrix is similarly transformed to a triangular shape that is almost diagonal.
• In the Schur normal form a matrix is unitary similarly transformed into a triangular matrix. The QR method calculates these numerically.

literature

• Gerd Fischer : Linear Algebra. (An introduction for first-year students). 13th revised edition. Vieweg, Braunschweig et al. 2002, ISBN 3-528-97217-3 .