In linear algebra, a diagonal matrix is a particular type of square matrix. The main-diagonal may contain any real values, but the elements that do not coincide with the main-diagonal must all be zeros.

In particular, given any ring ${\displaystyle R}$ with additive identity given by ${\displaystyle 0}$, a matrix ${\displaystyle A \in R^{n\times n} }$ (an ${\displaystyle n}$-by-${\displaystyle n}$ matrix whose entries are elements of ${\displaystyle R}$) is said to be a diagonal matrix if for any ${\displaystyle j}$ and ${\displaystyle k}$ such that ${\displaystyle j \ne k}$, ${\displaystyle A_{j,k}=0}$, where ${\displaystyle A_{j,k}}$ denotes the ${\displaystyle j, k}$-th entry.

An identity matrix is a special case of the diagonal matrix.

A matrix is considered diagonalizable if it is similar to a diagonal matrix. The process of finding the similar diagonal matrix is called diagonalization.

Diagonal matrices are very useful computationally, since

${\displaystyle \begin{bmatrix} a & 0 & 0\\ 0 & b & 0\\ 0 & 0 & c\end{bmatrix}^n = \begin{bmatrix} a^n & 0 & 0\\ 0 & b^n & 0\\ 0 & 0 & c^n \end{bmatrix}}$

${\displaystyle \begin{bmatrix} a & 0 & 0\\ 0 & b & 0\\ 0 & 0 & c\end{bmatrix} \begin{bmatrix} x & 0 & 0\\ 0 & y & 0\\ 0 & 0 & z \end{bmatrix} = \begin{bmatrix} ax & 0 & 0\\ 0 & by & 0\\ 0 & 0 & cz \end{bmatrix}}$

${\displaystyle \begin{bmatrix} a & 0 & 0\\ 0 & b & 0\\ 0 & 0 & c\end{bmatrix} + \begin{bmatrix} x & 0 & 0\\ 0 & y & 0\\ 0 & 0 & z \end{bmatrix} = \begin{bmatrix} a+x & 0 & 0\\ 0 & b+y & 0\\ 0 & 0 & c+z \end{bmatrix}}$

${\displaystyle \begin{bmatrix} a & 0 & 0\\ 0 & b & 0\\ 0 & 0 & c\end{bmatrix}^{-1} = \begin{bmatrix} \tfrac{1}{a} & 0 & 0\\ 0 & \tfrac{1}{b} & 0\\ 0 & 0 & \tfrac{1}{c} \end{bmatrix}}$

${\displaystyle \begin{vmatrix} a&0&0\\ 0&b&0\\ 0&0&c \end{vmatrix} =abc }$