A diagonal matrix is a special type of matrix in which the entries outside the main diagonal are all zero. In other words, a diagonal matrix has non-zero elements only on its diagonal, which runs from the top left to the bottom right of the matrix.
For example, consider the following 3×3 diagonal matrix:
D = | 5 0 0 |
| 0 9 0 |
| 0 0 4 |
In this matrix, the elements 5, 9, and 4 are on the diagonal, while all other entries are zero. This is a characteristic feature of diagonal matrices.
Diagonal matrices are significant in various mathematical contexts, especially in linear algebra. One of the main advantages of diagonal matrices is that they are easy to work with when it comes to operations like matrix multiplication and finding eigenvalues. For instance, if you multiply a diagonal matrix by a vector, you only need to multiply each element on the diagonal by the corresponding element in the vector.
Additionally, the properties of a diagonal matrix allow for straightforward calculations of powers of the matrix. For example, if we wish to compute the square of our diagonal matrix D:
D² = | 5² 0 0 |
| 0 9² 0 |
| 0 0 4² |
= | 25 0 0 |
| 0 81 0 |
| 0 0 16 |
As seen in this example, squaring a diagonal matrix results in another diagonal matrix with the squares of the original diagonal elements. This property of diagonal matrices plays a crucial role in simplifying many linear algebra problems.