How do you compute the commutator of two matrices?

Essential linear algebra for quantum computing

Source: Wikimedia Commons

In my last post, I shared five questions to help you get really good at the linear algebra needed for quantum computing. Let’s dive in and solve the fourth one step by step.

Question 4

Given two matrices A and B:

• Compute the commutator [A,B] = AB − BA

• Check if A and B commute.

Solution 4

A commutator tells us whether the order of multiplying two matrices matters.

For two matrices A and B, it is defined as:

[A,B] = AB − BA

If the result is zero, or AB = BA, we say that A and B commute.

Let’s first calculate AB as follows:

\(AB= \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}\)

Next, we calculate BA as follows:

\(BA= \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}\)

The commutator is calculated as follows:

\([A,B] = AB-BA = \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix} - \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 2\\ -2 & 0 \end{pmatrix}\)

Since [A,B] ≠ 0, A and B do not commute.

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