diagonal matrix

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

$\operatorname{diag}(a_{1},\ldots,a_{n})=\begin{pmatrix}a_{1}&0&0&\cdots&0\\ 0&a_{2}&0&\cdots&0\\ 0&0&a_{3}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\\ 0&0&0&&a_{n}\end{pmatrix}.$

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 $A B$ 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 (http://planetmath.org/TheoremForNormalTriangularMatrices)).

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

$\big{(}\operatorname{diag}(a_{1},\ldots,a_{n})\big{)}^{-1}=\operatorname{diag}% (1/a_{1},\ldots,1/a_{n}).$

4.

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

The matrix exponential of a diagonal matrix is

$e^{\operatorname{diag}(a_{1},\ldots,a_{n})}=\operatorname{diag}(e^{a_{1}},% \ldots,e^{a_{n}}).$

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

$f(\operatorname{diag}(a_{11},a_{22},...,a_{nn}))=\operatorname{diag}(f(a_{11})% ,f(a_{22}),...,f(a_{nn}))$

Remarks

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

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