Triangular matrix in the context of Matrix multiplication

⭐ Core Definition: Triangular matrix

In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero.

Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix L and an upper triangular matrix U if and only if all its leading principal minors are non-zero.

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Triangular matrix in the context of Determinant

In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix $A$ is commonly denoted $det(A)$ , $det A$ , or $| A |$ . Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse.

The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries.

View the full Wikipedia page for Determinant

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Triangular matrix in the context of Matrix multiplication

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⭐ Core Definition: Triangular matrix

Triangular matrix in the context of Determinant