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Triangular matrices

Created over 8 years ago, updated 10 days ago

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.