43. Triangular Matrix

A square matrix in which all entries on one side of the main diagonal are zero.

43.1. Upper triangular ()

All entries below the main diagonal are zero: for .

43.2. Lower triangular ()

All entries above the main diagonal are zero: for .

43.3. Properties (apply to both upper and lower)

• Determinant: product of diagonal entries

• Trace: sum of diagonal entries
• Eigenvalues: the diagonal entries themselves
• Sum / product / inverse of triangular matrices is triangular (same orientation)
• Transpose flips orientation: is upper triangular

43.4. Why they matter

Triangular systems are easy to solve by back substitution (upper) or forward substitution (lower):

Solve from the bottom row up — each row has one new unknown.

43.5. Where they show up

• Row Echelon Form: upper triangular (after Gaussian elimination)
• LU Decomposition: — lower × upper triangular
• Cholesky Decomposition: for symmetric positive-definite
• QR Decomposition: where is upper triangular