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Generalizations: 17.2 Row Echelon Form

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Upper-Triangular Matrix

Let $U \in R^{n \times n}$ be a given square matrix. We say that $U$ is an upper-triangular if $u_{i k} = 0$ for all $i > k$ . Upper-triangular matrices take the form

U = [\begin{matrix} u_{11} & u_{12} & \dots & u_{1 n} \\ 0 & u_{22} & ⋱ & U_{2 n} \\ ⋮ & ⋱ & ⋱ & ⋮ \\ 0 & \dots & 0 & u_{n n} \end{matrix}]

The upper-triangular entries of a matrix $A$ are all entries on or above the main diagonal. Thus, we say that element $a_{i k}$ is a upper-triangular entry if and only if $i \leq k$ .

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5e91140120203329ae2ab974f6f69323a1d35949: $I_{n} = [\begin{matrix} 1 & 0 & \dots & 0 \\ 0 & 1 & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & 1 \end{matrix}]$
03eb0ce7e91b85f44a7db2cfdff93cc5b71bfbbe: $[\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} & a_{46} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} & a_{56} \\ a_{61} & a_{62} & a_{63} & a_{64} & a_{65} & a_{66} \end{matrix}]$
2ca0bf9a1ed80d80f413a177c95f1e7e1e170d0b: $[\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} & a_{46} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} & a_{56} \\ a_{61} & a_{62} & a_{63} & a_{64} & a_{65} & a_{66} \end{matrix}]$
376bc28ac2eb8c3d2cc49152918b541f9172db4e: $[\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} & a_{46} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} & a_{56} \\ a_{61} & a_{62} & a_{63} & a_{64} & a_{65} & a_{66} \end{matrix}]$
79faa457317a00c9b18ba0ed1a2fa93e8241d467: $[\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} & a_{46} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} & a_{56} \\ a_{61} & a_{62} & a_{63} & a_{64} & a_{65} & a_{66} \end{matrix}]$
9e5e9c1128ca168e0329e9107ff78420f83067d5: $[\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} & a_{46} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} & a_{56} \\ a_{61} & a_{62} & a_{63} & a_{64} & a_{65} & a_{66} \end{matrix}]$

The strictly lower-triangular entries of a matrix $A$ are all entries $a_{i k}$ with $i < k$ .

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