Definition: A square matrix $A$ is:
- Upper triangular ($U$) if all entries below the main diagonal are zero: $a_{ij} = 0$ for $i > j$
- Lower triangular ($L$) if all entries above the main diagonal are zero: $a_{ij} = 0$ for $i < j$
- Strictly triangular if all diagonal entries are also zero
Example (upper triangular):
$$U = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{pmatrix}$$
Properties:
- The product of triangular matrices is triangular
- A triangular matrix is invertible iff all diagonal entries are nonzero
- The inverse of a triangular matrix is triangular
- The determinant is the product of diagonal entries
- Eigenvalues are the diagonal entries
Triangular matrices are important because many matrix factorizations (LU, Cholesky, QR) produce triangular factors, and solving triangular systems is computationally efficient via forward/backward substitution.
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- Definition of an upper triangular matrixDefinition. An $n \times n$ matrix $A = (a {ij})$ is called upper triangular if all entries below the main diagonal...
- Definition of a lower triangular matrixDefinition. An $n \times n$ matrix $A = (a {ij})$ is called lower triangular if all entries above the main diagonal...
- The product of upper/lower triangular matrices is upper/lower triangular.Theorem. The product of upper triangular matrices is upper triangular, and the product of lower triangular matrices...
- The inverse of an invertible upper/lower triangular matrix is upper/lower triangular.Theorem. If $A$ is an invertible upper triangular matrix, then $A^{-1}$ is also upper triangular. Similarly, if $A$...
- Every complex square matrix is unitarily similar to an upper triangular matrixTheorem (Schur)., Every square matrix over $\mathbb{C}$ is similar to an upper triangular matrix. That is, for any $A...
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