Triangular Matrix

43. Triangular Matrix

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

All entries below the main diagonal are zero: $𝑈_{𝑖 𝑗} = 0$ for $𝑖 > 𝑗$ .

𝑈 = [\begin{matrix} 𝑢_{11} & 𝑢_{12} & 𝑢_{13} \\ 0 & 𝑢_{22} & 𝑢_{23} \\ 0 & 0 & 𝑢_{33} \end{matrix}]

All entries above the main diagonal are zero: $𝐿_{𝑖 𝑗} = 0$ for $𝑖 < 𝑗$ .

𝐿 = [\begin{matrix} 𝑙_{11} & 0 & 0 \\ 𝑙_{21} & 𝑙_{22} & 0 \\ 𝑙_{31} & 𝑙_{32} & 𝑙_{33} \end{matrix}]

det (𝐿) = \prod_{𝑖 = 1}^{𝑛} 𝑙_{𝑖 𝑖}

Trace: sum of diagonal entries
Eigenvalues: the diagonal entries themselves
Sum / product / inverse of triangular matrices is triangular (same orientation)
Transpose flips orientation: $𝐿^{𝑇}$ is upper triangular

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

[\begin{matrix} 2 & 3 & 1 \\ 0 & 1 & - 1 \\ 0 & 0 & 4 \end{matrix}] \vec{𝑥} = [\begin{matrix} 7 \\ 2 \\ 8 \end{matrix}] \Rightarrow 𝑥_{3} = 2, 𝑥_{2} = 4, 𝑥_{1} = - 2

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