diagonal matrix
Definition Let $A$ be a square matrix^{} (with entries in any field). If all offdiagonal 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 ${a}_{1},{a}_{2},\mathrm{\dots},{a}_{n}$, then we denote the corresponding diagonal matrix by
$$\mathrm{diag}({a}_{1},\mathrm{\dots},{a}_{n})=\left(\begin{array}{ccccc}\hfill {a}_{1}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{\cdots}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {a}_{2}\hfill & \hfill 0\hfill & \hfill \mathrm{\cdots}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {a}_{3}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill 0\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \hfill & \hfill {a}_{n}\hfill \end{array}\right).$$ 
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 (http://planetmath.org/TheoremForNormalTriangularMatrices)).

3.
The eigenvalues^{} of a diagonal matrix $A=\mathrm{diag}({a}_{1},\mathrm{\dots},{a}_{n})$ are ${a}_{1},\mathrm{\dots},{a}_{n}$. Corresponding eigenvectors^{} are the standard unit vectors in ${\mathbb{R}}^{n}$. For the determinant^{}, we have $detA={a}_{1}{a}_{2}\mathrm{\cdots}{a}_{n}$, so $A$ is invertible^{} if and only if all ${a}_{i}$ are nonzero. Then the inverse is given by
$${\left(\mathrm{diag}({a}_{1},\mathrm{\dots},{a}_{n})\right)}^{1}=\mathrm{diag}(1/{a}_{1},\mathrm{\dots},1/{a}_{n}).$$ 
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
$${e}^{\mathrm{diag}({a}_{1},\mathrm{\dots},{a}_{n})}=\mathrm{diag}({e}^{{a}_{1}},\mathrm{\dots},{e}^{{a}_{n}}).$$
More generally, every analytic function^{} of a diagonal matrix can be computed entrywise, i.e.:
$$f(\mathrm{diag}({a}_{11},{a}_{22},\mathrm{\dots},{a}_{nn}))=\mathrm{diag}(f({a}_{11}),f({a}_{22}),\mathrm{\dots},f({a}_{nn}))$$ 
Remarks
Diagonal matrices are also sometimes called quasiscalar 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  20130322 13:43:32 
Last modified on  20130322 13:43:32 
Owner  rspuzio (6075) 
Last modified by  rspuzio (6075) 
Numerical id  12 
Author  rspuzio (6075) 
Entry type  Definition 
Classification  msc 1500 
Classification  msc 15A57 
Synonym  quasiscalar matrix 
Synonym  quasiscalar matrices 
Synonym  diagonal matrices 
Related topic  DiagonalizationLinearAlgebra 