A diagonal matrix is a square matrix where all elements outside the main diagonal are zero. For example:
[ 2 0 0 ]
[ 0 -5 0 ]
[ 0 0 3 ]Explanation:
- Square Matrix: A matrix with the same number of rows and columns.
- Main Diagonal: The elements from the top-left corner to the bottom-right corner. In the example above, the main diagonal consists of 2, -5, and 3.
- Zero Elements: All elements that are not on the main diagonal are zero.
Another Example:
The following is also a diagonal matrix:
[ 7 0 ]
[ 0 1 ]Key Properties of Diagonal Matrices:
- Diagonal matrices are easy to work with in many matrix operations.
- The determinant of a diagonal matrix is the product of the elements on the main diagonal. For example, in the first matrix shown above, the determinant would be (2) (-5) (3) = -30.
- The inverse of a diagonal matrix is another diagonal matrix, where each element on the main diagonal is the reciprocal of the corresponding element in the original matrix (provided none of the diagonal elements are zero).
