Special Matrices: Identity and Unitaries

The identity matrix is a very important matrix that comes up frequently in all types of mathematical applications.

It is a square matrix with all $1$s along the diagonal and $0$s everywhere else.

We denote it by $I_n$ where $n$ denotes the dimension of the matrix. Sometimes we omit the $n$ if it's clear from the context what the dimensions of $I$ are.

For example, the $2 \times 2$ identity matrix is:

$$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.$$

The $3 \times 3$ identity matrix is:

$$I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}.$$

In general, the $n \times n$ matrix is given by:

$$I_n = \begin{bmatrix} 1 & 0 & 0 & \dots & 0 \\ 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \end{bmatrix}.$$

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