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 $$ \diag(a_1,\ldots, a_n) = \begin{pmatrix} a_{1} & 0 & 0 & \cdots & 0 \\ 0 & a_{2} & 0 & \cdots & 0 \\ 0 & 0 & a_{3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \\ 0 & 0 & 0 & & a_{n} \end{pmatrix}. $$

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 $AB$ 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 $A=\diag(a_1,\ldots, a_n)$ are $a_1, \ldots, a_n$ Corresponding eigenvectors are the standard unit vectors in $\sR^n$ 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 $$ \big( \diag(a_1,\ldots, a_n)\big)^{-1} = \diag(1/a_1, \ldots, 1/a_n). $$
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 $$ e^{\diag(a_1,\ldots, a_n)} = \diag(e^{a_1}, \ldots, e^{a_n}). $$

Remarks

Bibliography

1

H. Eves, Elementary Matrix Theory, Dover publications, 1980.

2

Wikipedia, diagonal matrix.