Jordan Block

In the mathematical discipline of matrix theory, a Jordan block over a ring 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 \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 \lambda and is indicated as J_{\lambda,n}. Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix; using either the \oplus or the “\mathrm{diag}” symbol, the (n_1+\ldots+n_r) \times (n_1+\ldots+n_r) block diagonal square matrix consisting of r diagonal blocks, where the first is J_{\lambda_1,n_1}, the second is J_{\lambda_2,n_2}, \ldots, the r-th is J_{\lambda_r,n_r}, can be compactly indicated as J_{\lambda_1,n_1}\oplus \ldots \oplus J_{\lambda_r,n_r} or \mathrm{diag}\left(J_{\lambda_1,n_1}, \ldots, J_{\lambda_r,n_r}\right), respectively. For example the matrix is a 10\times 10 Jordan matrix with a 3\times 3 block with eigenvalue 0, two 2\times 2 blocks with eigenvalue the imaginary unit i, and a 3\times 3 block with eigenvalue 7. Its Jordan-block structure can also be written as either J_{0,3}\oplus J_{i,2}\oplus J_{i,2}\oplus J_{7,3} or \mathrm{diag}\left(J_{0,3},J_{i,2},J_{i,2},J_{7,3}\right).