|
|
|
|
diagonal matrix
|
(Definition)
|
|
|
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
- The identity matrix and zero matrix are diagonal matrices. Also, any
matrix is a diagonal matrix.
- A matrix
is a diagonal matrix if and only if is both an upper and lower triangular matrix.
- 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.
- A square matrix is diagonal if and only if it is triangular and normal (see this page).
- 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
- If
is a diagonal matrix, then the adjugate of is also a diagonal matrix.
- The matrix exponential of a diagonal matrix is
More generally, every analytic function of a diagonal matrix can be computed entrywise, i.e.:
Diagonal matrices are also sometimes called quasi-scalar matrices [1].
- 1
- H. Eves, Elementary Matrix Theory, Dover publications, 1980.
- 2
- Wikipedia, diagonal matrix.
|
"diagonal matrix" is owned by rspuzio. [ full author list (2) | owner history (2) ]
|
|
(view preamble)
See Also: diagonalization
| Other names: |
quasi-scalar matrix, quasi-scalar matrices, diagonal matrices |
This object's parent.
|
|
Cross-references: analytic function, matrix exponential, adjugate, inverse, invertible, determinant, unit vectors, eigenvalues, normal matrices, complex, real, order, lower triangular matrix, matrix, zero matrix, identity matrix, diagonal, off-diagonal entries, field, square matrix
There are 45 references to this entry.
This is version 9 of diagonal matrix, born on 2003-06-28, modified 2006-05-25.
Object id is 4411, canonical name is DiagonalMatrix.
Accessed 33108 times total.
Classification:
| AMS MSC: | 15-00 (Linear and multilinear algebra; matrix theory :: General reference works ) | | | 15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices ) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|