Jordan block
From Free net encyclopedia
In the mathematical discipline of matrix theory, a Jordan block is a matrix which is composed of zeros everywhere except the diagonal, which is filled with a certain number, and the superdiagonal, which is filled with 1s.
- <math>\begin{pmatrix}
\lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots& \vdots & \vdots \\ 0 & 0 & 0 & \lambda & 1 \\ 0 & 0 & 0 & 0 & \lambda \\\end{pmatrix}</math>
Any Jordan block is thus specified by its dimension n and its eigenvalue <math>\lambda</math> and is indicated as <math>J_{\lambda,n}</math>. Any Block_matrix#Block_diagonal_matrices whose blocks are Jordan blocks is called a Jordan matrix; using either the <math>\oplus</math> or the “<math>\mbox{diag}</math>” symbol, the <math>(l+m+n)\times (l+m+n)</math> block diagonal square matrix whose first diagonal block is <math>J_{\alpha,l}</math>, whose second diagonal block is <math>J_{\beta,m}</math> and whose third diagonal block is <math>J_{\gamma,n}</math> is compactly indicated as <math>J_{\alpha,l}\oplus J_{\beta,m}\oplus J_{\gamma,n}</math> or <math>\mbox{diag}\left(J_{\alpha,l}, J_{\beta,m}, J_{\gamma,n}\right)</math>, respectively.
