Háromszögmátrixnak

Háromszögmátrix, or triangular matrix in English, is a type of square matrix in linear algebra. It is characterized by having all entries above the main diagonal equal to zero (lower triangular matrix) or all entries below the main diagonal equal to zero (upper triangular matrix). The main diagonal consists of elements where the row index equals the column index.

A lower triangular matrix $L$ has entries $l_{ij} = 0$ for all $i < j$. An upper triangular matrix $U$ has entries $u_{ij} = 0$ for all $i > j$. A matrix that is both lower and upper triangular is a diagonal matrix, where all off-diagonal entries are zero.

Triangular matrices are important in linear algebra due to their simplifying properties. For instance, the determinant of a triangular matrix is simply the product of its diagonal entries. Solving systems of linear equations represented by triangular matrices is also significantly easier. For a lower triangular system, variables can be solved sequentially from top to bottom using forward substitution. For an upper triangular system, variables are solved from bottom to top using backward substitution.

The decomposition of a matrix into triangular factors, such as LU decomposition, is a fundamental technique in numerical analysis and scientific computing, used for solving linear systems, inverting matrices, and calculating determinants efficiently.