Diagonal matrix
Definition and Properties
A Diagonal matrix is a type of matrix in which the entries outside the main diagonal are all zero. The term is often used to refer to square matrices. The main diagonal itself may consist of either zeros, negative numbers, or positive numbers. The diagonal entries themselves are known as the principal elements or diagonal elements.
A diagonal matrix is often denoted by the symbol D or diag(a1, a2, ..., an), where a1, a2, ..., an are the diagonal elements. The size of a diagonal matrix is determined by the number of rows (or columns) it has. For example, a 3x3 diagonal matrix has three rows and three columns.
Characteristics of Diagonal Matrices
Diagonal matrices have several unique characteristics that distinguish them from other types of matrices. These include:
- Commutativity with other matrices: If A is any matrix and D is a diagonal matrix of the same size, then the product AD is equal to DA. This is a property that is not generally true for non-diagonal matrices.
- Simplicity of multiplication: When multiplying a diagonal matrix by a scalar, the operation is performed element-wise on the diagonal elements. Similarly, when multiplying two diagonal matrices together, the result is another diagonal matrix with elements that are the products of the corresponding elements in the original matrices.
- Ease of inversion: A diagonal matrix is invertible if and only if all of its diagonal elements are non-zero. The inverse of a diagonal matrix is another diagonal matrix with reciprocal elements on the diagonal.
Applications of Diagonal Matrices
Diagonal matrices are used in a variety of applications in mathematics and engineering. These include:
- Eigenvalues and eigenvectors: The eigenvalues of a diagonal matrix are simply its diagonal elements. This makes diagonal matrices especially useful in the study of eigenvalues and eigenvectors, which are fundamental concepts in linear algebra and many of its applications.
- Solving linear systems: Diagonal matrices are often used in numerical methods for solving systems of linear equations. Because of the simplicity of multiplication and inversion, algorithms that involve diagonal matrices can often be more efficient than those that involve general matrices.
- Matrix exponentiation: The process of raising a matrix to a power, known as matrix exponentiation, is particularly simple for diagonal matrices. The exponentiation of a diagonal matrix results in another diagonal matrix with elements that are the powers of the corresponding elements in the original matrix.
See Also
