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[parent] diagonal matrix (Definition)

Definition Let $ A$ be a square matrix (with entries in any field). If all off-diagonal entries of $ A$ are zero, then $ A$ is a diagonal matrix.

From the definition, we see that an $ n\times n$ diagonal matrix is completely determined by the $ n$ entries on the diagonal; all other entries are zero. If the diagonal entries are $ a_1, a_2, \ldots, a_n$, then we denote the corresponding diagonal matrix by

$\displaystyle \operatorname{diag}(a_1,\ldots, a_n) = \begin{pmatrix} a_{1} & 0 ... ...\ \vdots & \vdots & \vdots & \ddots & \ 0 & 0 & 0 & & a_{n} \end{pmatrix}. $

Examples

  1. The identity matrix and zero matrix are diagonal matrices. Also, any $ 1\times 1$ matrix is a diagonal matrix.
  2. A matrix $ A$ is a diagonal matrix if and only if $ A$ is both an upper and lower triangular matrix.

Properties

  1. If $ A$ and $ B$ are diagonal matrices of same order, then $ A+B$ and $ AB$ are again a diagonal matrix. Further, diagonal matrices commute, i.e., $ AB=BA$. 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).
  3. The eigenvalues of a diagonal matrix $ A=\operatorname{diag}(a_1,\ldots, a_n)$ are $ a_1, \ldots, a_n$. Corresponding eigenvectors are the standard unit vectors in $ \mathbb{R}^n$. For the determinant, we have $ \det A = a_1 a_2 \cdots a_n$, so $ A$ is invertible if and only if all $ a_i$ are non-zero. Then the inverse is given by
    $\displaystyle \big( \operatorname{diag}(a_1,\ldots, a_n)\big)^{-1} = \operatorname{diag}(1/a_1, \ldots, 1/a_n). $
  4. If $ A$ is a diagonal matrix, then the adjugate of $ A$ is also a diagonal matrix.
  5. The matrix exponential of a diagonal matrix is
    $\displaystyle e^{\operatorname{diag}(a_1,\ldots, a_n)} = \operatorname{diag}(e^{a_1}, \ldots, e^{a_n}). $
More generally, every analytic function of a diagonal matrix can be computed entrywise, i.e.:
$\displaystyle f(\operatorname{diag}(a_{11},a_{22},...,a_{nn}))= \operatorname{diag}(f(a_{11}),f(a_{22}),...,f(a_{nn})) $

Remarks

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

Bibliography

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) ]
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See Also: diagonalization

Other names:  quasi-scalar matrix, quasi-scalar matrices, diagonal matrices

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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 )
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