Identity Matrix
An identity matrix is a square matrix where all elements along its main diagonal are 1, and all other elements are 0.
Often referred to as a unit matrix or identical matrix, it's a fundamental concept in mathematics, particularly in linear algebra.
$$ I_n = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$
This matrix is commonly designated by "I" or "In", where "n" is the number of rows (or columns) the matrix houses.
To illustrate, let's consider a 4x4 identity matrix, where n is 4 because the matrix has 4 rows and 4 columns.
It's important to note that while every identity matrix is a square matrix, the converse doesn't hold true. In other words, a square matrix need not necessarily be an identity matrix.
Let's delve into a practical scenario.
Ponder over an identity matrix of dimension 3, denoted as I3. This 3x3 matrix looks like this:
$$ I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} $$
Here, the primary diagonal comprises of 1s while all the other elements are zeros.
Properties of Identity Matrices
Moving on to the properties of identity matrices, they harbor a range of unique and advantageous characteristics.
Perhaps the most noteworthy is their ability to retain the original matrix when multiplied with it.
Consider an arbitrary matrix M:
$$ M = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} $$
When matrix M undergoes multiplication by the identity matrix I2, the product is simply the original matrix M.
$$ M \cdot I_3 = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} $$
What this means is that multiplying a matrix by the identity matrix doesn't modify the original matrix, similar to how multiplying a number by 1 leaves the number unaffected.
This property bestows upon the identity matrix the status of the neutral element in the multiplicative ring of all n x n square matrices.
Note that, in this case, as they are both square matrices, the multiplication operation is commutative. Hence, whether you write M·I or I·M, the result is the same. $$ M \cdot I_3 = I_3 \cdot M = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} $$
Adding another feather to its cap, the identity matrix is also invertible. Its inverse is none other than the identity matrix itself.
These unique attributes make the identity matrix an indispensable tool in many mathematical procedures, particularly when it comes to solving systems of linear equations.
For instance, consider the identity matrix I2.$$ I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$ The inverse of I2 is identical to I2.$$ I_2^{-1} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$
So in essence, the identity matrix truly stands as an exceptional entity, playing an essential role in myriad mathematical operations and transformations.
