Main diagonal

Short description: Entries of a matrix for which the row and column indices are equal

In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix $\displaystyle{ A }$ is the list of entries $\displaystyle{ a_{i,j} }$ where $\displaystyle{ i = j }$. All off-diagonal elements are zero in a diagonal matrix. The following four matrices have their main diagonals indicated by red ones:

$\displaystyle{ \begin{bmatrix} \color{red}{1} & 0 & 0\\ 0 & \color{red}{1} & 0\\ 0 & 0 & \color{red}{1}\end{bmatrix} \qquad \begin{bmatrix} \color{red}{1} & 0 & 0 & 0 \\ 0 & \color{red}{1} & 0 & 0 \\ 0 & 0 & \color{red}{1} & 0 \end{bmatrix} \qquad \begin{bmatrix} \color{red}{1} & 0 & 0 \\ 0 & \color{red}{1} & 0 \\ 0 & 0 & \color{red}{1} \\ 0 & 0 & 0 \end{bmatrix} \qquad \begin{bmatrix} \color{red}{1} & 0 & 0 & 0 \\ 0 & \color{red}{1} & 0 & 0 \\ 0 & 0 &\color{red}{1} & 0 \\ 0 & 0 & 0 & \color{red}{1} \end{bmatrix} \qquad }$

Antidiagonal

The antidiagonal (sometimes counter diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal, or bad diagonal) of an order $\displaystyle{ N }$ square matrix $\displaystyle{ B }$ is the collection of entries $\displaystyle{ b_{i,j} }$ such that $\displaystyle{ i + j = N-1 }$ for all $\displaystyle{ 1 \leq i, j \leq N }$. That is, it runs from the top right corner to the bottom left corner.

$\displaystyle{ \begin{bmatrix} 0 & 0 & \color{red}{1}\\ 0 & \color{red}{1} & 0\\ \color{red}{1} & 0 & 0\end{bmatrix} }$