Main diagonal
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix [math]\displaystyle{ A }[/math] is the list of entries [math]\displaystyle{ a_{i,j} }[/math] where [math]\displaystyle{ i = j }[/math]. All off-diagonal elements are zero in a diagonal matrix. The following four matrices have their main diagonals indicated by red ones:
[math]\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 }[/math]
Antidiagonal
The antidiagonal (sometimes counter diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal, or bad diagonal) of an order [math]\displaystyle{ N }[/math] square matrix [math]\displaystyle{ B }[/math] is the collection of entries [math]\displaystyle{ b_{i,j} }[/math] such that [math]\displaystyle{ i + j = N-1 }[/math] for all [math]\displaystyle{ 1 \leq i, j \leq N }[/math]. That is, it runs from the top right corner to the bottom left corner.
- [math]\displaystyle{ \begin{bmatrix} 0 & 0 & \color{red}{1}\\ 0 & \color{red}{1} & 0\\ \color{red}{1} & 0 & 0\end{bmatrix} }[/math]
See also
- Trace
References
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