# Triangular matrices

Put content here**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.

# Parents

* Particular types of matrices