Question

Denoting the derivative of \(\mathbf{A}(t)\) by \(\dot{\boldsymbol{A}}\), show that $$ \frac{d}{d t}[\mathbf{A}, \mathbf{B}]=[\dot{\mathbf{A}}, \mathbf{B}]+[\mathbf{A}, \dot{\mathbf{B}}] . $$

Short answer

The derivative of the commutator with respective to time equals the sum of the commutator of derivative of A and B, and commutator of A and derivative of B.

Step-by-step solution

Understand the notation

The commutator of two matrices \( \mathbf{A} \) and \( \mathbf{B} \) is defined as \( [\mathbf{A}, \mathbf{B}] = \mathbf{AB} - \mathbf{BA} \). The derivative of a matrix \( \mathbf{A}(t) \) is denoted as \( \dot{\mathbf{A}} \) which is equivalent to \( d\mathbf{A}/dt \)

Differentiate the commutator

The derivative of the commutator \( [\mathbf{A}, \mathbf{B}] \) with respect to time is given by \( \frac{d}{dt} [\mathbf{A}, \mathbf{B}] = \frac{d}{dt} (\mathbf{AB} - \mathbf{BA}) \). By the product rule of differentiation, this equals \( \dot{\mathbf{A}}\mathbf{B} + \mathbf{A}\dot{\mathbf{B}} - \dot{\mathbf{A}}\mathbf{B} - \mathbf{A}\dot{\mathbf{B}} \). Which simplifies to \( [\dot{\mathbf{A}}, \mathbf{B}] + [\mathbf{A}, \dot{\mathbf{B}}]. \)

Form ,the conclusion

Thus we've shown that \( \frac{d}{dt}[\mathbf{A}, \mathbf{B}] = [\dot{\mathbf{A}}, \mathbf{B}] + [\mathbf{A}, \dot{\mathbf{B}}] \), which confirms the equation.

Key concepts

Commutator of Matrices

In mathematics, particularly in linear algebra, the commutator of matrices plays an essential role in understanding how matrix operations interact with each other. The commutator of two square matrices \textbf{A} and \textbf{B} is a measure of how non-commutative the matrix multiplication is. It is defined as \( [\textbf{A}, \textbf{B}] = \textbf{A}\textbf{B} - \textbf{B}\textbf{A} \).

When the commutator is the zero matrix, the matrices \textbf{A} and \textbf{B} are said to commute, meaning they can be multiplied in any order without affecting the result. The concept of commutation is fundamental in various branches of physics and mathematics, including quantum mechanics and group theory.

It's important to appreciate that the commutator encapsulates the difference that arises due to the order of multiplication. This can have profound implications for the behavior of systems described by such matrices, making the commutator a tool for analyzing changes and interactions.

Matrix Differentiation

Matrix differentiation is a process by which we find the derivative of a matrix-valued function with respect to a variable. This is analogous to taking the derivative of scalar functions in calculus but involves matrices. If we have a matrix function \( \textbf{A}(t) \) that depends on a variable \( t \), then the derivative of \( \textbf{A}(t) \) with respect to \( t \) is denoted by \( \dot{\textbf{A}} \) or \( \frac{d\textbf{A}}{dt} \).

The derivative of a matrix with respect to a variable is obtained by differentiating each element of the matrix individually with respect to that variable. This is especially useful in the context of linear dynamical systems where the system's evolution can be represented by matrix equations, and we need to calculate rates of change with respect to time or another variable.

Product Rule of Differentiation

The product rule is a fundamental principle in differential calculus used when taking the derivative of the product of two functions. It states that the derivative of a product is the derivative of the first function times the second function plus the first function times the derivative of the second function. Mathematically, if we have two differentiable functions \( f(t) \) and \( g(t) \), their product \( f(t)g(t) \) is differentiated as \( \frac{d}{dt}(f(t)g(t)) = f'(t)g(t) + f(t)g'(t) \).

This rule extends to matrix functions as well, enabling us to differentiate the product of two matrix functions with respect to a variable. In the context of matrix functions \( \textbf{A}(t) \) and \( \textbf{B}(t) \), the product rule allows us to correctly compute the derivative of their product, which is essential when dealing with systems expressed in matrix form.