Jordan matrix

In the mathematical discipline of matrix theory, a Jordan block over a ring ${\displaystyle R}$ (whose identities are the zero 0 and one 1) is a matrix composed of 0 elements everywhere except for the diagonal, which is filled with a fixed element ${\displaystyle \lambda \in R}$, and for the superdiagonal, which is composed of ones. The concept is named after Camille Jordan.

Every Jordan block is thus specified by its dimension n and its eigenvalue ${\displaystyle \lambda }$ and is indicated as ${\displaystyle J_{\lambda ,n}}$. Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix; using either the ${\displaystyle \oplus }$ or the “${\displaystyle \mathrm {diag} }$” symbol, the ${\displaystyle (l+m+n)\times (l+m+n)}$ block diagonal square matrix whose first diagonal block is ${\displaystyle J_{\alpha ,l}}$, whose second diagonal block is ${\displaystyle J_{\beta ,m}}$ and whose third diagonal block is ${\displaystyle J_{\gamma ,n}}$ is compactly indicated as ${\displaystyle J_{\alpha ,l}\oplus J_{\beta ,m}\oplus J_{\gamma ,n}}$ or ${\displaystyle \mathrm {diag} \left(J_{\alpha ,l},J_{\beta ,m},J_{\gamma ,n}\right)}$, respectively. For example the matrix

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