Jordan Matrix

Jordan Matrix

In the mathematical discipline of matrix theory, a Jordan block over a ring (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, and for the superdiagonal, which is composed of ones. The concept is named after Camille Jordan.

$\begin{pmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots& \ddots & \vdots \\ 0 & 0 & 0 & \lambda & 1 \\ 0 & 0 & 0 & 0 & \lambda \end{pmatrix}$

Every Jordan block is thus specified by its dimension n and its eigenvalue and is indicated as . Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix; using either the or the “” symbol, the block diagonal square matrix whose first diagonal block is, whose second diagonal block is and whose third diagonal block is is compactly indicated as or, respectively. For example the matrix

$J=\left(\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & i & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & i & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & i & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 \end{matrix}\right)$

is a Jordan matrix with a block with eigenvalue, two blocks with eigenvalue the imaginary unit and a block with eigenvalue 7. Its Jordan-block structure can also be written as either or .

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