diagonal matrix (original) (raw)

From the definition, we see that an n×n diagonal matrix is completely determined by the n entries on the diagonal; all other entries are zero. If the diagonal entries are a1,a2,…,an, then we denote the corresponding diagonal matrix by

Examples

2. 1. A matrix A is a diagonal matrix if and only if A is both an upper and lower triangular matrix.

Properties

1. 1. If A and B are diagonal matrices of same order, thenA+B and A⁢B are again a diagonal matrix. Further, diagonal matrices commute, i.e., A⁢B=B⁢A. It follows that real (and complex) diagonal matrices are normal matrices.
2. 1. A square matrix is diagonal if and only if it is triangular and normal (see this page (http://planetmath.org/TheoremForNormalTriangularMatrices)).
3. 1. The eigenvalues of a diagonal matrixA=diag⁡(a1,…,an) are a1,…,an. Corresponding eigenvectors are the standard unit vectors in ℝn. For the determinant, we have det⁡A=a1⁢a2⁢⋯⁢an, soA is invertible if and only if all ai are non-zero. Then the inverse is given by

      (diag⁡(a1,…,an))-1=diag⁡(1/a1,…,1/an).

4. 1. If A is a diagonal matrix, then the adjugate of A is also a diagonal matrix.

More generally, every analytic function of a diagonal matrix can be computed entrywise, i.e.:

Remarks

Diagonal matrices are also sometimes called quasi-scalar matrices [1].

References