Calculate The Following Commutator

In linear algebra, the commutator of two operators or matrices measures how much they fail to commute. This calculator helps you compute the commutator of two given matrices or operators.

What is a Commutator?

The commutator of two operators A and B, denoted [A, B], is defined as the difference between the product of A and B and the product of B and A:

[A, B] = AB - BA

If the commutator is zero, the operators commute, meaning AB = BA. Otherwise, the operators do not commute.

Commutators are important in quantum mechanics, where they describe how quantum states evolve under different operators. In group theory, commutators help identify the commutator subgroup.

Commutator Formula

The general formula for the commutator of two matrices A and B is:

[A, B] = AB - BA

Where:

A and B are square matrices of the same size
AB is the matrix product of A and B
BA is the matrix product of B and A

The result is another matrix of the same size as A and B.

How to Calculate a Commutator

Step 1: Input the Matrices

Enter the two matrices A and B in the calculator. The matrices must be of the same size.

Step 2: Compute the Products

Calculate the matrix products AB and BA.

Step 3: Subtract the Products

Subtract BA from AB to get the commutator [A, B].

Step 4: Interpret the Result

If the result is a zero matrix, the operators commute. Otherwise, they do not commute.

Examples

Example 1: 2x2 Matrices

Let A = [1 2; 3 4] and B = [0 1; 1 0].

Compute AB = [1*0+2*1 1*1+2*0; 3*0+4*1 3*1+4*0] = [2 1; 4 3].

Compute BA = [0*1+1*3 0*2+1*4; 1*1+0*3 1*2+0*4] = [3 4; 1 2].

The commutator [A, B] = AB - BA = [2-3 1-4; 4-1 3-2] = [-1 -3; 3 1].

Example 2: Commuting Matrices

Let A = [1 0; 0 -1] and B = [0 1; 1 0].

Compute AB = [1*0+0*1 1*1+0*0; 0*0+(-1)*1 0*1+(-1)*0] = [0 1; -1 0].

Compute BA = [0*1+1*0 0*0+1*(-1); 1*1+0*0 1*0+0*(-1)] = [0 -1; 1 0].

The commutator [A, B] = AB - BA = [0-0 1-(-1); -1-1 0-0] = [0 2; -2 0].

Applications

Commutators have important applications in:

Quantum mechanics: Describing how quantum states evolve under different operators
Group theory: Identifying the commutator subgroup
Lie algebras: Understanding the structure of Lie groups
Physics: Analyzing symmetries and conservation laws

FAQ

What is the difference between a commutator and an anticommutator?

A commutator is defined as [A, B] = AB - BA, while an anticommutator is defined as {A, B} = AB + BA. The commutator measures non-commutativity, while the anticommutator measures commutativity.

When is the commutator zero?

The commutator [A, B] is zero if and only if the operators A and B commute, meaning AB = BA.

Can the commutator be negative?

The commutator itself is a matrix, not a scalar, so it doesn't have a sign. However, individual elements of the commutator matrix can be negative.