Main diagonal

Definition

The main diagonal of a square matrix consists of the elements that run from the top left to the bottom right corner. These elements have equal row and column indices.

5 Must Know Facts For Your Next Test

1. In a $3 \times 3$ matrix, the main diagonal elements are at positions $(1,1)$, $(2,2)$, and $(3,3)$.
2. For a matrix $A$, the main diagonal elements are denoted as $a_{ii}$ where $i$ is the index.
3. In Gaussian Elimination, pivot elements are often located on or moved to the main diagonal.
4. The identity matrix is characterized by having all ones on its main diagonal and zeros elsewhere.
5. Zero or very small values on the main diagonal can indicate potential numerical instability in solving systems of equations.

Related terms

Pivot Element:

An element used as a reference point for row operations during Gaussian Elimination.

$LU$ Decomposition: $A$ factorization of a matrix into a lower triangular matrix ($L$) and an upper triangular matrix ($U$).

Identity Matrix:

$A$ square matrix with ones on its main diagonal and zeros elsewhere.