Types: 8.6 Identity Matrix
Examples: N/A
Constructions: N/A
Generalizations: N/A

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Diagonal Matrix

Let $D \in R^{n \times n}$ be a given, square matrix. We say that $D$ is diagonal if $d_{i k} = 0$ for all $i \neq k$ . Diagonal matrices take the form

D = [\begin{matrix} d_{11} & 0 & \dots & 0 \\ 0 & d_{22} & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & d_{n n} \end{matrix}]

In this case, the entries on the main diagonal $d_{i i}$ are any real numbers and are not necessarily zero.

The main diagonal entries of $A$ are elements with equal row and column indices. We say $a_{i k}$ is on the main diagonal of $A$ if $i = k$ . The main diagonal of $A$ is the set of all diagonal entries of $A$ .

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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 non-diagonal entries of $A$ are all entries that are not in the main diagonal (in this case, the green strip).

Remark. These are central to 4 The Standard Eigenvalue Problem.
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