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In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. Because matrix equations with triangular matrices are easy to solve they are very important in numerical analysis. The LU decomposition gives an algorithm to decompose any invertible matrix A into a normed lower triangle matrix L and an upper triangle matrix U. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...
For the square matrix section, see square matrix. ...
In linear algebra, the main diagonal of a square matrix is the diagonal which runs from the top left corner to the bottom right corner. ...
For the square matrix section, see square matrix. ...
Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ...
In linear algebra, a LU decomposition, or LUP decomposition is a matrix decomposition of a matrix into a lower triangular matrix L, an upper-triangular matrix U, and a permutation matrix P. This decomposition is used in numerical analysis to solve systems of linear equations. ...
In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...
Definition
A matrix is called lower triangular matrix or left triangular matrix, and analogously a matrix of the form is called upper triangular matrix or right triangular matrix.
A triangular matrix with zero entries on the main diagonal is strictly upper or lower triangular. All strictly triangular matrices are nilpotent. In linear algebra, the main diagonal of a square matrix is the diagonal which runs from the top left corner to the bottom right corner. ...
In mathematics, a nilpotent matrix is a square matrix that is nilpotent. ...
If the entries on the main diagonal are 1, the matrix is termed unit upper/lower or normed upper/lower triangular. If, in addition, all the off-diagonal entries are zero except for the entries in one column, then the matrix is atomic upper/lower triangular; such a matrix is also called a Gauss (transformation) matrix. So an atomic lower triangular matrix is of the form
The inverse of an atomic triangular matrix is again atomic triangular. Indeed, we have i.e. the off-diagonal entries are replaced by their opposites.
Notes A matrix which is simultaneously upper and lower triangular is diagonal. The identity matrix is the only matrix which is both normed upper and lower triangular. In linear algebra, a diagonal matrix is a square matrix in which only the entries in the main diagonal are non-zero. ...
In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...
The transpose of a upper triangular matrix is a lower triangular matrix and vice versa. The determinant of a triangular matrix equals the product of the diagonal entries. In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...
In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
The variable L is commonly used for lower triangular matrix, standing for lower/left, while the variable U or R is commonly used for upper triangular matrix, standing for upper/right.
Generally, operations can be performed on triangular matrices within half the time.
Generalizations The product of two upper triangular matrices is upper triangular, so the set of upper triangular matrices forms an algebra. Algebras of upper triangular matrices have a natural generalization in functional analysis which yields nest algebras on Hilbert spaces. In mathematics, an associative algebra is a vector space (or more generally module) which also allows the multiplication of vectors in a distributive and associative manner. ...
Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...
In functional analysis, nest algebras are a class of operator algebras which generalise the upper-triangular matrix algebras to a Hilbert space context. ...
In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...
The set of invertible triangular matrices form a group, and is a subgroup of all invertible matrices. The set of 2 by 2 triangular matrices is called the parabolic subgroup; 3 by 3 and larger normed triangular matrices form the Heisenberg group. Both are examples of a Borel subgroup. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
Möbius transformations should not be confused with the Möbius transform. ...
In mathematics, the Heisenberg group is a group of 3×3 upper triangular matrices of the form Elements a,b,c can be taken from some (arbitrary) commutative ring. ...
In mathematics, a Borel subgroup (named after Armand Borel) of an algebraic group G is a maximal solvable subgroup. ...
Examples The matrix is upper triangular and is lower triangular.
The matrix
is atomic lower triangular and its inverse is
Application A matrix equation in the form or is very easy to solve. The matrix equation Lx = b can be written as a system of linear equations which can be solved by the following recursive relation -
A matrix equation with an upper triangular matrix U can be solved in an analogous way.
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