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:

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

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.

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

See also

• Trace

References

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

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This page was last edited on 18 January 2024, at 07:17