diagonal matrix
Definition
Let be a square matrix (with entries in any field).
If all off-diagonal entries of are zero, then is a
diagonal matrix
.
From the definition, we see that an diagonal matrix is completely determined by the entries on the diagonal; all other entries are zero. If the diagonal entries are , then we denote the corresponding diagonal matrix by
Examples
-
1.
The identity matrix
and zero matrix
are diagonal matrices. Also, any matrix is a diagonal matrix.
-
2.
A matrix is a diagonal matrix if and only if is both an upper and lower triangular matrix
.
Properties
-
1.
If and are diagonal matrices of same order, then and are again a diagonal matrix. Further, diagonal matrices commute, i.e., . It follows that real (and complex) diagonal matrices are normal matrices
.
-
2.
A square matrix is diagonal if and only if it is triangular and normal (see this page (http://planetmath.org/TheoremForNormalTriangularMatrices)).
-
3.
The eigenvalues
of a diagonal matrix are . Corresponding eigenvectors
are the standard unit vectors in . For the determinant
, we have , so is invertible
if and only if all are non-zero. Then the inverse is given by
-
4.
If is a diagonal matrix, then the adjugate of is also a diagonal matrix.
-
5.
The matrix exponential
of a diagonal matrix is
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
-
1
H. Eves,
Elementary Matrix
Theory, Dover publications, 1980.
- 2 Wikipedia, http://www.wikipedia.org/wiki/Diagonal_matrixdiagonal matrix.
Title | diagonal matrix |
---|---|
Canonical name | DiagonalMatrix |
Date of creation | 2013-03-22 13:43:32 |
Last modified on | 2013-03-22 13:43:32 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 12 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 15-00 |
Classification | msc 15A57 |
Synonym | quasi-scalar matrix |
Synonym | quasi-scalar matrices |
Synonym | diagonal matrices |
Related topic | DiagonalizationLinearAlgebra |