Main diagonal

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:

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

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.

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

(*) Secondary (as well as trailing, minor and off) diagonals very often also mean the (a.k.a. k-th) diagonals parallel to the main or principal diagonals, i.e., ${\displaystyle A_{i,\,i\pm k}}$ for some nonzero k =1, 2, 3, ... More generally and universally, the off diagonal elements of a matrix are all elements not on the main diagonal, i.e., with distinct indices i ≠ j.

• Trace

• Weisstein, Eric W. "Main diagonal". MathWorld.