Diagonally dominant matrix
A diagonally dominant matrix is a special type of matrix where the numbers along the diagonal (the line of numbers from the top left corner to the bottom right corner) are themselves bigger than the sum of the other numbers in their respective rows or columns. This means that the diagonal elements are dominant or more important than the other elements in their row or column.
For example, let’s take a matrix with the following numbers:
- 2 6 3
- 1 5 2
- 4 8 10
The diagonal elements are 2, 5, and 10. We can see that each of these numbers is bigger than the sum of the other numbers in their respective rows or columns. For instance, in the first row, 2 is bigger than 6+3=9. In the second row, 5 is bigger than 1+2=3. Finally, in the last row, 10 is bigger than 4+8=12.
This special property of diagonally dominant matrices is very important in the study of linear algebra, as it can help us solve systems of equations more easily and consistently. It also means that we can use certain numerical methods for solving these equations that work particularly well with diagonally dominant matrices, such as the famous Gauss-Seidel method.
For example, let’s take a matrix with the following numbers:
- 2 6 3
- 1 5 2
- 4 8 10
The diagonal elements are 2, 5, and 10. We can see that each of these numbers is bigger than the sum of the other numbers in their respective rows or columns. For instance, in the first row, 2 is bigger than 6+3=9. In the second row, 5 is bigger than 1+2=3. Finally, in the last row, 10 is bigger than 4+8=12.
This special property of diagonally dominant matrices is very important in the study of linear algebra, as it can help us solve systems of equations more easily and consistently. It also means that we can use certain numerical methods for solving these equations that work particularly well with diagonally dominant matrices, such as the famous Gauss-Seidel method.