Linear Algebra/Matrix Equation

Diagonal Matrix

A diagonal matrix, $A$, is a square matrix in which the entries outside of the main diagonal are zero. The main diagonal of a square matrix consists of the entries which run from the top left corner to the bottom right corner.

In the example below the main diagonal are $a_{11},a_{22},...,a_{nn}$

$A=\left[\begin{array}{cccc}{a}_{11}& 0& \cdots & 0\\ 0& a_{22}& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & a_{nn}\end{array}\right]$

Identity Matrix

The identity matrix, with a size of n, is an n-by-n square matrix with ones on the main diagonal and zeros elsewhere. It is commonly denoted as $I_n$, or simply by I if the size is immaterial or can be easily determined by the context.

$I_1=[1]I_2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]I_3=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]I_n=\left[\begin{array}{cccc}1& 0& \cdots & 0\\ 0& 1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 1\end{array}\right]$

The most important property of the identity matrix is that, when multiplied by another matrix, A, the result will be A

$A{I}_n=A$ and $I_{n}A=A$.