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An upper triangular matrix system with nonzero diagonal is solved from the last row upward. Once $x_{i+1},\dots ,x_n$ are known, use $x_i=(b_i-{\sum }_{j>i}{a}_{ij}{x}_j)/a_{ii}$. The nonzero diagonal guarantees a unique solution; this is the final solve after a QR factorization.

Every real symmetric positive-definite matrix has a unique factorization $A=L{L}^T$ in which $L$ is a lower triangular matrix with positive diagonal entries.

Each QR decomposition $A_k=Q_k{R}_k$ givesInduction therefore yields the first Accumulated QR factorization identity:

We next prove the corresponding factorization of a matrix power. The case $k=0$ is just $A=Q_0{R}_0$. Ifthen the similarity formula above impliesConsequently,The product ${\overline{Q}}_k$ is orthogonal, and the product ${\overline{R}}_k$ is upper triangular. Henceis a QR decomposition.

Using Euler's formula,ThusOrdering the modes as $(-1,0,1)$, let $a=1/2+i/4$. ThenConsequentlyThis matrix is triangular, so its eigenvalues are its diagonal entries:

The upper unitriangular group consists of upper triangular matrices with every diagonal entry equal to one. Over ${\mathbb{F}}_p$ it has order $p^{n(n-1)/2}$ and is a Sylow subgroup of $G{L}_n({\mathbb{F}}_p)$.