Identity Transformation and Matrix

The identity matrix is the matrix that does nothing when multiplied to any matrix.

More clearly, if $A$ is any matrix and $I$ is the identity matrix, $$AI=IA=A$$ First, note that $I$ must be a square matrix, or else the matrix $A$ would not stay the same. Also, it is easy to see that $I$ will not always be the same matrix; it changes depending on the size of the matrix, but we will see that it changes in a very predictable pattern.

It turns out that the identity matrix is the square matrix with $1$ in every position along the main diagonal and $0$ in every other position in the matrix. This is quite easy to prove, although a bit tedious.

For example, the $3\times 3$ identity matrix looks like this: $$I_3=\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$$ You are encouraged to verify that the product of this matrix with any other matrix will not change that matrix, and that this holds regardless of the order in which you multiply.

Now what does this mean in the context of transformations? The identity matrix also has an associated transformation, and as you may have guessed, it's the identity transformation! In other words, for a transformation $F$ and the identity $I$, $$F\circ I=I\circ F=F$$ The identity transformation keeps everything exactly as it is, which means that it has many pleasing properties. For example, it is also an isometry, since distances are obviously preserved.

The identity matrix also brings up the notion of inverse matrices. An inverse matrix is one that, when multiplied with the given matrix, yields the identity matrix, regardless of order.