Question

Verify that $$ \frac{d}{d t} \mathbf{H}^{3}=\left(\frac{d \mathbf{H}}{d t}\right) \mathbf{H}^{2}+\mathbf{H}\left(\frac{d \mathbf{H}}{d t}\right) \mathbf{H}+\mathbf{H}^{2}\left(\frac{d \mathbf{H}}{d t}\right) . $$

Short answer

Having followed the computations, it can be concluded that indeed \(\frac{d}{d t} \mathbf{H}^{3} = \mathbf{H}^{2} \frac{d \mathbf{H}}{d t} + \mathbf{H} \frac{d \mathbf{H}}{d t} \mathbf{H} + \frac{d \mathbf{H}}{d t} \mathbf{H}^{2}\), thereby verifying the equality provided in the exercise.

Step-by-step solution

Compute the derivative of \(\mathbf{H}^{3}\)

We start by computing the derivative of \(\mathbf{H}^{3}\). Using the chain rule, this can be broken down as \(\frac{d}{d t} \mathbf{H}^{3} = 3 \mathbf{H}^{2} \frac{d \mathbf{H}}{d t}\).

Use the Distributive Property of Matrix Multiplication

The right-hand side can be considered as a sum of three terms, all with \(\frac{d \mathbf{H}}{d t}\) as a factor in different positions. Remember that we can't generally swap the factors in a product, however in this case we rely on the distributive property to expand \(3 \mathbf{H}^{2} \frac{d \mathbf{H}}{d t}\) into \(\mathbf{H}^{2} \frac{d \mathbf{H}}{d t} + \mathbf{H} \frac{d \mathbf{H}}{d t} \mathbf{H} + \frac{d \mathbf{H}}{d t} \mathbf{H}^{2}\).

Comparison and Conclusion

Now we can see that the derivation result corresponds to the right-hand side of the given equation. Thus, we can assert that the given formula is correct.

Key concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra involving two matrices. The product of two matrices is calculated by taking the dot products of the rows of the first matrix with the columns of the second matrix.
This means you need to be careful about the order of multiplication because matrix multiplication is not commutative. In other words, \( AB eq BA \) generally.
When multiplying matrices:

• The number of columns in the first matrix must match the number of rows in the second matrix.
• The resulting product will have dimensions equal to the number of rows of the first matrix by the number of columns of the second.

Understanding how matrices multiply is essential for working with transformations, such as squaring a matrix or performing higher-order operations like cubing.

Chain Rule

The chain rule is a powerful tool in calculus that helps in finding the derivative of composite functions. This rule is adaptable to matrix calculus, facilitating the differentiation of matrix-valued functions.
In the given problem, the chain rule allows us to differentiate \( \mathbf{H}^{3} \). By breaking down the power of the matrix, we use the chain rule to express that the derivative of \( \mathbf{H}^{n} \) involves multiplying the derivative of the matrix \( \mathbf{H} \) by instances of the matrix and its powers:

• Differentiate the outer function, treating the inner matrix as a variable.
• Then multiply this by the derivative of the inner matrix function.

For example, for the cube of a matrix, this involves calculating \( 3 \mathbf{H}^{2} \frac{d \mathbf{H}}{d t} \), ensuring that each term is multiplied appropriately, respecting matrix multiplication rules.

Derivative of Matrix Functions

The derivative of matrix functions refers to the process of finding how a matrix-valued function changes with respect to a variable. This is analogous to finding the derivative of scalar functions but involves matrix operations.
When dealing with matrices, special rules apply due to the non-commutative nature of matrix multiplication. In the exercise, we see this when breaking the derivative into individual terms:

• Each term in the derivative has a different placement of \( \frac{d \mathbf{H}}{d t} \) because distributing it differently respects the original positions of matrices.
• This leads to an expression like \( \mathbf{H}^{2} \frac{d \mathbf{H}}{d t} + \mathbf{H} \frac{d \mathbf{H}}{d t} \mathbf{H} + \mathbf{H}^{2} \left(\frac{d \mathbf{H}}{d t}\right) \), which accounts for the intricate relationships of matrices during differentiation.

By understanding these rules, one can tackle complex problems involving matrix derivatives, leading to insights into systems governed by linear transformations.