Matrix Diagonals¶

Definition¶

For an $m\times n$ matrix $\boldsymbol{A}=(a_{ij})$ $(a_{ij})$ :

the main diagonal consists of the entries $a_{ii}$ with $i\le\min(m,n)$ $min (m, n)$ ;
the anti-diagonal of a square matrix ( $m =$ $m=n$ $n$ ) consists of the entries $a_{n+1-i,i}$ ;
the off-diagonal entries are all entries $a_{ij}$ with $i\neq j$ $j$ .

Interpretation¶

The main diagonal runs from the top-left entry downwards to the right, through the entries whose row and column index agree. The anti-diagonal of a square matrix runs from bottom-left to top-right; the off-diagonal entries are all the others.

A 3 by 4 matrix with a line through the entries a11, a22 and a33, its main diagonal.

In the $3\times4$ matrix shown, the main diagonal (shaded) is $a_{11},a_{22},a_{33}$ . It stops after $\min(3,4)=3$ entries, so a non-square matrix has a main diagonal too, but it does not end in the bottom-right corner.

Diagonal matrices have zeros everywhere off the main diagonal and triangular matrices on one side of it. The transpose mirrors a matrix in it, so the diagonal entries stay in place.

Key Results¶

Diagonal entries are central in definitions of diagonal matrices, triangular matrices, and the trace.
A square matrix is diagonal exactly when all off-diagonal entries are zero.

Examples¶

In $\begin{pmatrix}1&2\\3&4\end{pmatrix}$ , the main diagonal is $(1,4)$ and the anti-diagonal is $(3,2)$ .
In $\begin{pmatrix}1&0\\0&4\end{pmatrix}$ , all off-diagonal entries are zero.

Prerequisites¶

matrix

Sources¶

Math II: Linear Algebra · Week 4

Aliases: matrix diagonal, leading diagonal Keywords: principal diagonal, main diagonal, anti-diagonal, antidiagonal, off-diagonal, diagonal entries

Last updated: 20 October 2026 03:02