Question

Show that \(\mathbf{U}=\exp \mathbf{A}\) is unitary if \(\mathbf{A}\) is anti- hermitian. Furthermore, if A commutes with \(\mathbf{A}^{\dagger}\), then \(\exp \mathbf{A}\) is unitary. Hint: Use Proposition \(4.2 .4\) on \(\mathbf{U U}^{\dagger}=\mathbf{1}\) and \(\mathbf{U}^{\dagger} \mathbf{U}=\mathbf{1}\)

Short answer

Using the properties of anti-hermitian matrices and proposition 4.2.4, it can be shown that a matrix \( \mathbf{U} \) derived from the exponential of an anti-hermitian matrix \( \mathbf{A} \), is unitary.

Step-by-step solution

Verify anti-hermitian nature of A

An anti-hermitian matrix has the property that its Hermitian conjugate (the conjugate transpose) equals its negative. In mathematical terms, if \( \mathbf{A} \) is anti-hermitian, then \( \mathbf{A}^{\dagger} = - \mathbf{A} \). Verify that this is true for the given matrix \( \mathbf{A} \).

Apply the property of anti-hermitian to U

Express U in terms of A i.e. \( \mathbf{U} = \exp {\mathbf{A}} \). Since \( \mathbf{A} \) is anti-hermitian, we can use the property from step 1 to express the Hermitian conjugate of U as \( \mathbf{U}^{\dagger} = \exp {(\mathbf{A}^{\dagger})} = \exp {(- \mathbf{A})} \). This gives us the expressions for both U and its Hermitian conjugate.

Show U is Unitary

To prove U is unitary, we need to show that \( \mathbf{U} \mathbf{U}^{\dagger} = \mathbf{1} \) and \( \mathbf{U}^{\dagger} \mathbf{U} = \mathbf{1} \). Using the expressions from step 2, we can write these as \( \exp {\mathbf{A}} \exp {(- \mathbf{A})} = \mathbf{1} \) and \( \exp {(- \mathbf{A})} \exp {\mathbf{A}} = \mathbf{1} \). If A commutes with \( \mathbf{A}^{\dagger} \), then we can simplify these expressions and we conclude that \( \mathbf{U} \) is indeed unitary.

Key concepts

Anti-Hermitian Matrix

An Anti-Hermitian Matrix, in the world of linear algebra, is somewhat like the bizarro twin of a Hermitian matrix. It's a complex square matrix that, when you transpose it and take the complex conjugate of all its entries (a process known as taking the Hermitian conjugate), what you get is the negative of the original matrix. Mathematically speaking, if you have a matrix \( \mathbf{A} \) that's anti-Hermitian, then \( \mathbf{A}^\dagger = -\mathbf{A} \).

Why should you care about anti-Hermitian matrices? Well, in physics, they pop up all the time, especially in quantum mechanics where observable quantities are represented by Hermitian matrices, and the generators of unitary transformations, which are key in describing symmetries and dynamics, are anti-Hermitian. They have this cool property that if you crank out the matrix exponential (don't worry, we'll get to what that means in a bit), you'll end up with a unitary matrix, which is a matrix that's all about preserving the length of vectors—a truly handy property for keeping things normalized in quantum states.

Hermitian Conjugate

The Hermitian Conjugate of a matrix, also dubbed the adjoint of a matrix in some math circles, is like giving your matrix a thorough mirror-and-invert treatment. You start with a complex square matrix, flip it over its diagonal (that's the transpose), and then replace every element with its complex conjugate. The notation for the Hermitian conjugate of a matrix \( \mathbf{A} \) is \( \mathbf{A}^\dagger \).

This whole process might seem like a mathematical stunt, but it's a cornerstone concept when dealing with unitary matrices. Remember, a unitary matrix is one where the matrix times its Hermitian conjugate gives you the identity matrix (the mathematical equivalent of a reset button). In the exercise, we take advantage of this property to transform an anti-Hermitian matrix into a unitary one. So, mastering the art of the Hermitian conjugate is not just a brain teaser; it's your ticket to understanding deeper principles of linear transformations, particularly in spaces where complex numbers rule the roost.

Matrix Exponential

Last but not least, let's talk about the Matrix Exponential. You've heard of exponentials with numbers, but with whole matrices? Yep, that's a thing! When mathematicians talk about \( \exp(\mathbf{A}) \), where \( \mathbf{A} \) is a matrix, they're referring to a special function that springs from series expansion. In simpler terms, it's like taking \( e \) (the irrational but lovable constant of natural logarithms) to the power of each element of your matrix—but with a truckload of matrix multiplication mixed in.

The matrix exponential has a magical property: it can morph an anti-Hermitian matrix into a unitary one. That's a big deal because unitary matrices are the guardians of vector length in complex vector spaces, a property that's gold in fields like quantum mechanics. When you apply the matrix exponential to an anti-Hermitian matrix, it ensures that the resulting matrix can perform transformations without messing up the 'size' of whatever it's transforming. This concept underlies the exercise we're dealing with, showcasing that sweet spot where abstract algebra meets practical, reality-based applications.