Question

Show that for any \(\alpha, \beta \in \mathbb{R}\) and any \(\mathbf{H} \in \operatorname{End}(\mathcal{V})\), we have $$ e^{\alpha \mathbf{H}} e^{\beta \mathbf{H}}=e^{(\alpha+\beta) \mathbf{H}} . $$

Short answer

In conclusion, it's shown that for any \(\alpha, \beta \in \mathbb{R}\) and any \(\mathbf{H} \in \operatorname{End}(\mathcal{V})\), the expression \(e^{\alpha \mathbf{H}} e^{\beta \mathbf{H}}=e^{(\alpha+\beta) \mathbf{H}}\) indeed holds true.

Step-by-step solution

- Indicate the matrices

Let's begin by representing \(e^{\alpha H}\) and \(e^{\beta H}\) as infinite series. We have \(e^{\alpha H} = I + \alpha H + \frac{\alpha^2 H^2}{2!} + \frac{\alpha^3 H^3}{3!} + ... \) and \(e^{\beta H} = I + \beta H + \frac{\beta^2 H^2}{2!} + \frac{\beta^3 H^3}{3!} + ... \), where \(H\) is the matrix. Note that \(H\) is a linear transformation from a vector space to itself, and thus, the matrices commute.

- Compute the multiplication

Let's multiply \(e^{\alpha H}\) and \(e^{\beta H}\). Note that as the matrices commute (i.e \(H^i H^j = H^j H^i\)), each term in \(e^{\alpha H}\) can be expanded and rearranged to get a term in \(e^{(\alpha+ \beta) H}\). For instance, \(\frac{\alpha^2 H^2}{2!}\cdot\frac{\beta^3 H^3}{3!} = \frac{\alpha^2 \beta^3 H^5}{2!3!}\), which can be rearranged to be a term in the expansion of \(e^{(\alpha+ \beta) H}\).

- Applying Binomial theorem

We know that \((\alpha + \beta )^n = \sum_{i=0}^{n}{n \choose i} \alpha^i \beta^{(n-i)}\). Thus when we rearrange each term in \(e^{\alpha H} \cdot e^{\beta H}\), we get terms in \(e^{(\alpha+ \beta) H}\), which represent all the possible combinations of \(\alpha\) and \(\beta\) powers, as per the binomial theorem. We can extend this argument to all terms and thus conclude that \(e^{\alpha H}\cdot e^{\beta H}= e^{(\alpha+ \beta) H}\).

Key concepts

Linear Transformations

Linear transformations play a pivotal role in understanding matrix exponentiation. They are mappings from a vector space to itself that preserve vector addition and scalar multiplication. If you have a matrix, let's call it \( \mathbf{H} \), it can be seen as a linear transformation because when it acts on a vector, it provides a new vector in the same space.

One key aspect of linear transformations is that they can be represented using matrices. This matrix representation helps us perform operations like exponentiation in a systematic way.
For example, if you apply the transformation twice, it is equivalent to multiplying the matrix \( \mathbf{H} \) by itself, denoted as \( \mathbf{H}^2 \).

• Linear transformations respect the operations of addition and scalar multiplication.
• They map vectors in a vector space back into the same space.

Understanding these transformations is crucial, especially when dealing with expressions like \( e^\alpha \mathbf{H} \) as seen in matrix exponentiation.

Commutative Matrices

Commutative matrices are pairs of matrices that can be multiplied in any order with the same result. This property is essential for solving the exercise about the equality \( e^{\alpha \mathbf{H}} e^{\beta \mathbf{H}} = e^{(\alpha+\beta) \mathbf{H}} \).

Typically, matrices do not commute; \( \mathbf{A} \mathbf{B} eq \mathbf{B} \mathbf{A} \) in general. However, when dealing with the same matrix \( \mathbf{H} \,\), multiplied to different powers as in the given exercise, they do commute. This means:

• \( \mathbf{H}^i \mathbf{H}^j = \mathbf{H}^j \mathbf{H}^i \)

This property simplifies the calculation significantly since you can reorder terms freely during multiplication. For example, when expanding \( e^{\alpha \mathbf{H}} \cdot e^{\beta \mathbf{H}} \), the commutativity ensures that all combinations of terms match with those in \( e^{(\alpha + \beta) \mathbf{H}} \).

Binomial Theorem

The binomial theorem is a crucial tool that helps simplify expressions involving powers, crucial for understanding how the matrices in the original exercise relate to one another. The theorem states that:

\[(\alpha + \beta)^n = \sum_{i=0}^{n} {n \choose i} \alpha^i \beta^{n-i}\]

In the context of matrix exponentiation, it allows us to systematically expand and rearrange the series into smaller, more manageable parts. When matrices commute, as with \( \mathbf{H} \) in our exercise, this theorem helps express the product \( e^{\alpha \mathbf{H}} \cdot e^{\beta \mathbf{H}} \) as \( e^{(\alpha + \beta) \mathbf{H}} \).

The theorem provides a framework to multiply terms by reorganizing the added powers of \( \alpha \) and \( \beta \) and fitting them into the exponential form needed.

Vector Spaces

Vector spaces provide the foundational structure on which linear transformations, like those discussed in matrix exponentiation, operate. A vector space is a collection of vectors that can be added together and multiplied by scalars while staying within the same space.

The concepts of linear transformations and commutative matrices are inherently tied to the properties of vector spaces. These spaces are equipped with two operations: vector addition and scalar multiplication. They serve as the domain and range for linear transformations represented by matrices.

• Vectors in a vector space have both magnitude and direction.
• They can be subject to linear transformations.
• Operations within this space obey specific algebraic rules.

Understanding vector spaces helps in appreciating why we can represent transformations with matrices and why these matrices can behave differently under certain conditions.