Identity Matrices

A zero matrix of size $m\times n$, denoted by $\mathbf{0}_{m,n}$ is a matrix with all entries as $0$. Multiplying any valid matrix by a zero matrix results in a zero matrix as well.

An identity matrix is a square matrix (that is, it is of the size $m\times m$), when multiplied by another matrix results in the same original matrix. Multiplying a matrix by the identity matrix leaves it unchanged, illustrating a key property of matrix multiplication. It is usually denoted by $I_{m}$ if the matrix has size $m\times m$. If $\mathcal B$ is a matrix of size $m\times n$ which is multiplicable with $I$, then $I_m\cdot\mathcal B=\mathcal B\cdot I_n=B$.

You can think of identity matrix as $1$ (multiplicative identity) for real numbers. For real numbers $a\cdot1=1\cdot a=a$, similarly $A\cdot I=I\cdot A=A$.

Here's a little exercise with zero and identity matrices.

First, write the 2 by 2 identity matrix.

It should look like this.

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

Now, write the 5 by 5 identity matrix.

It should look like this.

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

Finally, write the 3 by 3 zero matrix.

It should look like this.

$\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$

(Videos) The Identity Matrix and the Zero Matrix

by slcmath@pc

Here's a good video by slcmath@pc showing you what identity matrices look like and showing how multiplying a matrix by an identity matrix is like multiplying a number by 1.

And here's slcmath@pc's video on the zero matrix.

(Video) Identity Matrix

by Don't Memorise

This video by Don't Memorise explains what an identity matrix is and how we use it.

Summary

In math, "identity" refers to something that gives us our original value. Think back to the Multiplicative Identity Property, where $1 \cdot 3=3$. Any number multiplied by $1$ gives us back our original number.

$$AI = A$$

Similarly, multiplying any matrix by the identity matrix will result in the original matrix. An identity matrix (or the unit matrix) is any square matrix where all of the terms in the leading diagonal are $1$, and the rest of the terms in the matrix are $0$.

A $2x2$ identity matrix looks like:

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

A $3x3$ identity matrix looks like:

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

Example: Find matrix $AI$.

$$A = \begin{bmatrix} 1&0 \\ -2&3 \\ \end{bmatrix}$$

We can see that this is a $2x2$ square matrix, so the associated identity matrix will be: $\begin{bmatrix} 1&0 \\0&1 \\ \end{bmatrix}$.

$$AI = \begin{bmatrix} 1&0 \\ -2&3 \\ \end{bmatrix} \cdot \begin{bmatrix} 1&0 \\0&1 \\ \end{bmatrix}$$

Both of these matrices are $2x2$ matrices, so the matrices are compatible, resulting in a $2x2$ resulting matrix.

$$AI = \begin{bmatrix} 1&0 \\ -2&3 \\ \end{bmatrix} \cdot \begin{bmatrix} 1&0 \\0&1 \\ \end{bmatrix} = \begin{bmatrix} 1&0 \\ -2&3 \\ \end{bmatrix}$$

We can see that $AI = A$. Let's look at it the other way.

$$IA = \begin{bmatrix} 1&0 \\0&1 \\ \end{bmatrix} \cdot \begin{bmatrix} 1&0 \\ -2&3 \\ \end{bmatrix}$$

Both of these matrices are $2x2$ matrices, so the matrices are compatible, resulting in a $2x2$ resulting matrix.

$$IA = \begin{bmatrix} 1&0 \\0&1 \\ \end{bmatrix} \cdot \begin{bmatrix} 1&0 \\ -2&3 \\ \end{bmatrix} = \begin{bmatrix} 1&0 \\ -2&3 \\ \end{bmatrix}$$

We can see that $IA=A$.

Now that we've learned the basics of the row and column matrix, we can move forward into more complicated types of matrix multiplication.

Identity Matrices

Identity matrices, $I$, are matrices where all entries are $0$, except for the diagonal. That means, for a given matrix, $A$, $AI=A$.

As we can see from the image, a $2x2$ identity matrix looks like: $\begin{bmatrix} 1&0 \\ 0&1 \\ \end{bmatrix}$. A $3x3$ identity matrix looks like: $\begin{bmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1 \\ \end{bmatrix}$.

The pattern is not within the rows and columns, but actually in the diagonal. The $1$s are all in the diagonal, starting at the top left.

So, for example, we will find that:

$$\begin{bmatrix} 5&5 \\ 3&4 \\ \end{bmatrix} \cdot \begin{bmatrix} 1&0 \\ 0&1 \\ \end{bmatrix} = \begin{bmatrix} 5&5 \\ 3&4 \\ \end{bmatrix}$$

Take It a Step Further

If you think about it, an identity matrix is equivalent to $1$. We will commonly see identity matrices in matrix equations, especially as a multiplicative inverse. You will find that the multiplicative inverse of matrix $A$ can be multiplied by the multiplicative inverse to get the identity matrix.