Triangular matrix
Definition
A Triangular matrix is a specific type of square matrix where all the entries above or below the main diagonal are zero. The name derives from the shape of the matrix, which resembles a triangle when visualized. Triangular matrices are a subset of the broader category of diagonal matrices, where all off-diagonal elements are zero.
Types of Triangular Matrices
There are two types of triangular matrices: upper triangular matrices and lower triangular matrices.
Upper Triangular Matrix
An Upper triangular matrix is a special type of square matrix where all the entries below the main diagonal are zero. The main diagonal itself and the elements above it may contain non-zero elements.
Lower Triangular Matrix
A Lower triangular matrix, on the other hand, is a square matrix where all the entries above the main diagonal are zero. The main diagonal and the elements below it may contain non-zero elements.
Properties of Triangular Matrices
Triangular matrices have several unique properties that make them useful in various mathematical and computational applications.
- The determinant of a triangular matrix is the product of its diagonal elements. This property simplifies the process of calculating the determinant, especially for large matrices.
- The sum and difference of two triangular matrices of the same type (both upper or both lower) is another triangular matrix of that type.
- The product of two upper (or lower) triangular matrices is an upper (or lower) triangular matrix.
- The inverse of an invertible upper (or lower) triangular matrix is an upper (or lower) triangular matrix.
- The eigenvalues of a triangular matrix are the entries of its main diagonal.
Applications of Triangular Matrices
Triangular matrices are used extensively in numerical linear algebra, particularly in the methods for solving systems of linear equations, eigenvalues, and matrix factorization.
Solving Linear Systems
One of the most common applications of triangular matrices is in the solution of linear systems of equations. This is typically achieved through a process known as Gaussian elimination or LU decomposition, which transforms the original system into an equivalent one where the coefficient matrix is upper or lower triangular.
Eigenvalues
The eigenvalues of a triangular matrix are simply the entries of its main diagonal. This property is useful in various computational algorithms, such as the QR algorithm for computing the eigenvalues of a matrix.
Matrix Factorization
Triangular matrices are also used in LU decomposition, a form of matrix factorization. In this process, a matrix is decomposed into the product of a lower triangular matrix and an upper triangular matrix. This decomposition is useful for solving linear systems, computing determinants, and performing other matrix operations.