Diagonal - Matrices

Matrices

In the case of a square matrix, the main or principal diagonal is the diagonal line of entries running from the top-left corner to the bottom-right corner. For a matrix with row index specified by and column index specified by, these would be entries with . For example, the identity matrix can be defined as having entries of 1 on the main diagonal and zeroes elsewhere:

$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

The top-right to bottom-left diagonal is sometimes described as the minor diagonal or antidiagonal. The off-diagonal entries are those not on the main diagonal. A diagonal matrix is one whose off-diagonal entries are all zero.

A superdiagonal entry is one that is directly above and to the right of the main diagonal. Just as diagonal entries are those with, the superdiagonal entries are those with . For example, the non-zero entries of the following matrix all lie in the superdiagonal:

$\begin{pmatrix} 0 & 2 & 0 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{pmatrix}$

Likewise, a subdiagonal entry is one that is directly below and to the left of the main diagonal, that is, an entry with . General matrix diagonals can be specified by an index measured relative to the main diagonal: the main diagonal has ; the superdiagonal has ; the subdiagonal has ; and in general, the -diagonal consists of the entries with .