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.
**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.