Main diagonal

Definition

The main diagonal of a matrix is the diagonal that runs from the top-left corner to the bottom-right corner. It consists of elements $a_{ii}$ where the row and column indices are equal.

5 Must Know Facts For Your Next Test

1. In a square matrix, the main diagonal contains $n$ elements where $n$ is the number of rows (or columns) in the matrix.
2. The elements on the main diagonal are crucial when determining if a matrix is invertible; if any element is zero, special consideration is required.
3. During Gaussian elimination, operations often focus on making all entries below (and above) the main diagonal zero.
4. For identity matrices, all entries on the main diagonal are 1, and all other entries are 0.
5. In certain types of matrices like diagonal matrices, only the main diagonal contains non-zero elements.

Review Questions

• What defines an element as part of the main diagonal in a matrix?
• How does the main diagonal relate to an identity matrix?
• Why is it important to focus on elements below and above the main diagonal during Gaussian elimination?

Related terms

Gaussian Elimination: A method for solving systems of linear equations by transforming a matrix into reduced row echelon form using row operations.

Diagonal Matrix: A type of square matrix in which all off-diagonal elements are zero and only elements on the main diagonal may be non-zero.

Identity Matrix: A square matrix with ones on its main diagonal and zeros elsewhere; it acts as a multiplicative identity in matrix multiplication.