Diagonal matrix

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

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

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

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

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

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