Main diagonal of a Square matrix
A straight path that connects the entries (or elements) whose row and column are the same in a square matrix is called the main diagonal of a square matrix.
Introduction
In a square matrix, the total number of elements in a row is equal to the total number of elements in a column. Hence, the first element of the first row and the last element of the last row can be connected by a straight path.
This path contains an element from each row. For example, the first element from the first row, second element from the second row, and so on. Hence, it is called the main diagonal of a square matrix. It is also called by the following four names.
- Principal diagonal of a square matrix
- Primary diagonal of a square matrix
- Leading diagonal of a square matrix
- Major diagonal of a square matrix
Let’s understand the concept of leading diagonal of a matrix $M$.
$M$ $\,=\,$ $\begin{bmatrix} \color{red} e_{11} & e_{12} & e_{13} & \cdots & e_{1n} \\ e_{21} & \color{red} e_{22} & e_{23} & \cdots & e_{2n} \\ e_{31} & e_{32} & \color{red} e_{33} & \cdots & e_{3n} \\ \vdots & \vdots & \vdots & \color{red} \ddots & \vdots \\ e_{n1} & e_{n2} & e_{n3} & \cdots & \color{red} e_{nn} \\ \end{bmatrix}$
The matrix $M$ contains $n$ rows and $n$ columns. Hence, the arrangement of elements forms a square matrix.
In the square matrix $M$, the elements $e_{11}$, $e_{22}$, $e_{33}$, $e_{44}$ and so on can be connected by a straight path diagonally. Hence, the diagonal path is called the main diagonal of the square matrix $M$.
Example
Let $P$ be a square matrix of order $4$. In this $4 \times 4$ square matrix, there are $16$ elements. The sixteen elements are arranged as four rows and four columns.
$P$ $\,=\,$ $\begin{bmatrix} \color{red} 1 & -3 & 5 & 2 \\ 6 & \color{red} 0 & 4 & 2 \\ -2 & 7 & \color{red} 3 & 9 \\ 5 & -3 & -5 & \color{red} 8\end{bmatrix}$
The elements $1$, $0$, $3$ and $8$ can be connected diagonally by a straight path. Hence, It is known as the major diagonal of the square matrix $P$.