A calculus of the absurd

18.1.6 The identity matrix

In the same way that we have the number \(1\) which is an “identity” (or “neutral”) element for multiplication in the real numbers 126126 And also the natural numbers, integers, rational and irrational numbers., we have an identity matrix. This is usually denoted with the letter \(I\) and by definition satisfies the property that for any \(m \times m\) matrix and any \(m \times m\) matrix \(A\), it is the case that

\begin{equation} AI = I = IA \end{equation}

We can define \(A\) as

\begin{equation} (A)_{ij} = \begin{cases} 1 \text { if } i = j \\ 0 \text { otherwise } \end {cases} \end{equation}

That is, a diagonal line of \(1\)s and \(0\) everywhere else. For example, for the dimension \(2 \times 2\), \(I\) is

\begin{equation} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end {pmatrix} \end{equation}

And for the \(3\times 3\) case, it is

\begin{equation} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {pmatrix} \end{equation}

The same is true for higher dimensions.