Identity Problems

Identity Matrix?! What on the good green Earth is an Identity Matrix?

The short answer is that it’s a square matrix with 1’s down the main diagonal and 0’s everywhere else. For example, a 3×3 identity matrix looks like this:

$\boldsymbol{\large{\begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}}}$

That’s not particularly useful to understanding what it’s all about though, so let’s have a closer look at it.

The idea of an identity element is an important concept in Abstract Algebra, which is the study of the algebraic structures of mathematics. To understand what an identity is in the context of matrices, first it’s probably best to look at something we’re a bit more familiar with – like regular old numbers:

$\boldsymbol{\large{10+0=10}}$

Adding 0 to 10 does not change the value. I still have 10. This is what an identity element does. When I apply an identity to something else, I don’t change the value of that thing. So 0 is an identity. But it is not the identity. Because:

$\boldsymbol{\large{10\times 0=0}}$

If I multiply 10 by 0 the result is definitely not 10. I’ve changed its value. So 0 is only an identity in the case of addition – it is the additive identity.

What about multiplication then? Is there something we can multiply a number by and always get an unchanged value? Of course there is:

$\boldsymbol{\large{10\times 1=10}}$

If I multiply any number by 1, I don’t change the value of that number. So 1 is the multiplicative identity.

An identity is specific to a particular operation on a particular set of objects. In the examples above, those objects are numbers and the operations are addition and multiplication.

So how does it work for matrices?

An identity matrix does the same thing for matrix multiplication that the number 1 does for number multiplication. For example:

$\boldsymbol{\large{\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}\times\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}=\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}}}$

If we multiply a matrix by the identity matrix, we don’t change the original matrix at all. All the rules of matrix multiplication still apply. In particular, the identity matrix needs to be of an appropriate size such that we can do the multiplication to begin with. I’m assuming here that you’re familiar with matrix multiplication itself. If not, then that’s a topic for another post.

An identity matrix is often represented by an I (capital i) along with a subscript indicating its size if required, so for example:

$\boldsymbol{\large{I_2=\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}\quad\quad\quad I_3=\begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}}}$

So for any matrix $\boldsymbol{A}$ and appropriately sized identity matrices $\boldsymbol{I}$

$\boldsymbol{\large{AI=A\quad\text{ and }\quad IA=A}}$

But what about adding matrices?

Great question. Yeah, the thing that we’re calling the identity matrix is actually a multiplicative identity matrix, right? It’s specific to the operation of matrix multiplication.

There is an additive identity for matrices as well. It would be a matrix (of an appropriate size) in which every element is 0. For example:

$\boldsymbol{\large{\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}+\begin{bmatrix}0 & 0 \\0 & 0\end{bmatrix}=\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}}}$

Most of the actual work we do with matrices involves multiplying them. Adding matrices is of, comparatively, little interest and so the additive identity is far less used. So I think (I don’t know) that’s why we commonly just refer to the multiplicative as the identity matrix.

That’s all for now.

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