An upper or lower triangular matrix has all entries on one side of the main diagonal equal to zero. Its eigenvalues, with algebraic multiplicity, are its diagonal entries.
An upper triangular matrix system with nonzero diagonal is solved from the last row upward. Once are known, use . The nonzero diagonal guarantees a unique solution; this is the final solve after a QR factorization.
An upper triangular matrix has zero entries below its main diagonal. Its determinant is the product of its diagonal entries, and an invertible upper triangular linear system is solved by backward substitution.
A square matrix is strictly upper triangular if every entry on or below its diagonal is zero. A product of such -by- matrices vanishes, so they form a nilpotent Lie algebra under the commutator. Multiplication by an upper triangular matrix preserves strict upper triangularity and zero trace.
A lower triangular matrix has zero entries above its main diagonal. Its determinant is the product of its diagonal entries, and an invertible lower triangular linear system is solved by forward substitution.
The algebra consists of the lower triangular by matrices over . Its Jacobson radical consists of the strictly lower triangular matrices, andThe simple left modules are one-dimensional, with a matrix acting through its th diagonal entry. For the left regular module,
A triangular linear system is solved by forward substitution for a lower triangular coefficient matrix or backward substitution for an upper triangular coefficient matrix, using operations for a dense by matrix.
Forward substitution solves a lower-triangular system from its first row downward, substituting each newly determined variable into the rows below.
Backward substitution solves an upper-triangular system from its last row upward.
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A triangular matrix is a special type of square matrix (a matrix with an equal number of rows and columns) that has specific characteristics regarding the placement of its non-zero elements. There are two main types of triangular matrices: 1. **Upper Triangular Matrix**: An upper triangular matrix is a square matrix where all the elements below the main diagonal are zero.
