Main diagonal

In linear algebra, the main diagonal (sometimes leading diagonal or major diagonal or primary diagonal or principal diagonal) of a matrix A is the collection of entries A_i,j where i is equal to j.

The main diagonal of a square matrix is the diagonal which runs from the top left corner to the bottom right corner. For example, the following matrix has 1s down its main diagonal:

$\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}.$

A square matrix like the above in which the entries outside the main diagonal are all zero is called a diagonal matrix. The sum of the entries on the main diagonal of a square matrix is known as the trace of that matrix.

The main diagonal of a rectangular matrix is the diagonal which runs from the top left corner and steps down and right, until the right edge or the bottom edge is reached.

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

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

The diagonal of a square matrix from the top right to the bottom left corner is called antidiagonal, counterdiagonal, secondary diagonal, or minor diagonal.

References

• Weisstein, Eric W., "Main diagonal" from MathWorld.
• Main diagonal in Mathwords