Main diagonal

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

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

Antidiagonal

See also: Anti-diagonal matrix

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

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

(*) 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., $A_{i,\,i\pm k}$ 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.

See also

• Trace

References

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