Triangular matrix

Triangular matrix - in linear algebra, a square matrix in which all elements below (or above) the main diagonal are equal to zero.

An example of an upper triangular matrix

Content

Basic Definitions

The upper triangular matrix (or upper triangular matrix ) is a square matrix $A$ ${\ displaystyle A}$ , in which all elements below the main diagonal are equal to zero: $a_{ij}=0$ ${\ displaystyle a_ {ij} = 0}$ at $i>j$ ${\ displaystyle i> j}$ ^[1] ^[2]

Lower Triangular Matrix (or Lower Triangular Matrix ) - Square Matrix $A$ ${\ displaystyle A}$ , in which all elements above the main diagonal are equal to zero: $a_{ij}=0$ ${\ displaystyle a_ {ij} = 0}$ at $i<j$ ${\ displaystyle i <j}$ ^[1] ^[2] .

Unit triangular matrix (top or bottom) - triangular matrix $A$ ${\ displaystyle A}$ , in which all elements on the main diagonal are equal to unity: $a_{jj}=1$ ${\ displaystyle a_ {jj} = 1}$ ^[3] .

The diagonal matrix is both the upper triangular and the lower triangular ^[4] .

Application

Triangular matrices are used primarily in solving systems of linear algebraic equations (SLAE). For example, the Gauss method for solving SLAEs is based on the following result ^[5] :

any matrix $A_{n\times n}$ ${\ displaystyle A_ {n \ times n}}$ by elementary transformations over strings and permutations of strings, we can lead to a triangular form.

Thus, the solution of the initial SLAE reduces to solving a system of linear equations with a triangular matrix of coefficients, which is not difficult.

There is a variant of this method (called the compact scheme of the Gauss method) based on the following results ^[6] :

any square matrix $A$ ${\ displaystyle A}$ with non-zero leading principal minors can be represented as the product of the lower triangular matrix $L$ ${\ displaystyle L}$ and upper triangular matrix $U$ ${\ displaystyle U}$ : $A=LU$ ${\ displaystyle A = LU}$ (see LU decomposition), and such a decomposition is unique if the diagonal elements of one of the two triangular matrices are fixed in advance - for example, you can require that $L$ ${\ displaystyle L}$ was unitriangular;
any non-degenerate square matrix $A$ ${\ displaystyle A}$ can be represented as follows: $PA=LU$ ${\ displaystyle PA = LU}$ where $P$ ${\ displaystyle P}$ - permutation matrix (selected in the process of constructing the decomposition) (see LUP decomposition).

Properties

The determinant of a triangular matrix is equal to the product of the elements of its main diagonal ^[7] (in particular, the determinant of a unitriangular matrix is equal to unity).
The set of nondegenerate upper triangular matrices of order n in multiplication with elements from the field k forms a group ^[4] , which is denoted by UT ( n , k ) or UT _n ( k ).
The set of nondegenerate lower triangular matrices of order n in multiplication with elements from the field k forms a group ^[4] , which is denoted by LT ( n , k ) or LT _n ( k ).
The set of upper unitriangular matrices with elements from the field k forms a subgroup UT _n ( k ) by multiplication, which is denoted by SUT ( n , k ) or SUT _n ( k ). A similar subgroup of lower unitriangular matrices is denoted by SLT ( n , k ) or SLT _n ( k ).
The set of all upper triangular matrices with elements from the associative ring k forms an algebra with respect to the operations of addition, multiplication by elements of the ring, and multiplication of matrices. A similar statement is true for lower triangular matrices.
The group UT _{n is} decidable , and its unitriangular subgroup SUT _{n is} nilpotent .

Notes

↑ ¹ ² Voevodin and Kuznetsov, 1984 , p. 27.
↑ ¹ ² Ikramov, 1991 , p. 9-10.
↑ Ikramov, 1991 , p. ten.
↑ ¹ ² ³ Gantmakher, 1988 , p. 27.
↑ Gantmakher, 1988 , p. 42-43.
↑ Voevodin and Kuznetsov, 1984 , p. 76, 174-175.
↑ Voevodin and Kuznetsov, 1984 , p. thirty.

Literature

Voevodin V.V. , Kuznetsov Yu.A. Matrices and calculations. - M .: Nauka , 1984. - 320 p.
Gantmakher F.R. Matrix Theory. 4th ed. - M .: Nauka , 1988 .-- 552 p. - ISBN 5-02-013722-7 .
Ikramov Kh. D. Asymmetric eigenvalue problem. Numerical methods. - M .: Nauka , 1991 .-- 240 p. - ISBN 5-02-014462-2 .

[_f42a5c574e34d46e-1] ¹ ² Voevodin and Kuznetsov, 1984 , p. 27.

[_4a179771dc98f400-2] ¹ ² Ikramov, 1991 , p. 9-10.

[_58ec58cef3d12c5d-3] Ikramov, 1991 , p. ten.

[_0e8092996f7967f3-4] ¹ ² ³ Gantmakher, 1988 , p. 27.

[_3523a8bd70e254a5-5] Gantmakher, 1988 , p. 42-43.

[_18165fdbcd4d10a7-6] Voevodin and Kuznetsov, 1984 , p. 76, 174-175.

[_f42a5b574e34d296-7] Voevodin and Kuznetsov, 1984 , p. thirty.