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Chapter 4: Problem 2

Prove that the general form of a \(2 \times 2\) unitary, unimodular matrix is $$ \mathbf{U}=\left(\begin{array}{rr} a & b \\ -b^{*} & a^{*} \end{array}\right) $$ with \(a^{*} a+b^{*} b=1\).

Short Answer

Expert verified
The matrix satisfies unitary and unimodular conditions when \(a^* a + b^* b = 1\).

Step by step solution

01

Understanding Unitary Matrices

A matrix is unitary if it satisfies the equation \( U^* U = I \), where \( U^* \) is the conjugate transpose of \( U \) and \( I \) is the identity matrix. For a \(2 \times 2\) matrix \( U = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the condition becomes \( \begin{pmatrix} a^* & c^* \ b^* & d^* \end{pmatrix} \begin{pmatrix} a & b \ c & d \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \).
02

Calculating the Conjugate Transpose

For the given matrix \( \mathbf{U}=\begin{pmatrix} a & b \-b^* & a^* \end{pmatrix} \), its conjugate transpose \( \mathbf{U}^* \) is \( \begin{pmatrix} a^* & -b \ b^* & a \end{pmatrix} \).
03

Checking the Unitarity Condition

Calculate \( \mathbf{U}^* \mathbf{U} \): \[ \mathbf{U}^* \mathbf{U} = \begin{pmatrix} a^* & -b \ b^* & a \end{pmatrix} \begin{pmatrix} a & b \ -b^* & a^* \end{pmatrix} = \begin{pmatrix} a^*a + b^*b & a^*b - b a^* \ b^*a - a b^* & b^*b + a^*a \end{pmatrix} \].Simplify to \( \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \) using the condition that \(a^* a + b^* b = 1\).
04

Verifying Unimodularity

A matrix is unimodular if its determinant is 1. Calculate the determinant of \( \mathbf{U} \):\[\det(\mathbf{U}) = a \cdot a^* - b(-b^*) = a a^* + b^*b.\]Using the condition \( a^* a + b^* b = 1 \), we confirm that \( \det(\mathbf{U}) = 1 \).
05

Conclusion

The given form \( \mathbf{U}=\begin{pmatrix} a & b \ -b^{*} & a^{*} \end{pmatrix} \) satisfies both the unitarity and unimodularity conditions when \( a^* a + b^* b = 1 \), proving it is a unitary, unimodular matrix.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unitarity
A unitary matrix is a special type of matrix with unique properties. In simple terms, a matrix is unitary if multiplying it by its conjugate transpose yields the identity matrix. This essentially means that when you apply a unitary matrix to any vector in the complex plane, the length of the vector remains unchanged. This is known as preserving the norm of the vector.

Here鈥檚 a basic rundown of what unitarity implies:
  • Unitarity ensures that the matrix preserves inner products and hence length and angles in vector operations.
  • For a unitary matrix \( U \), the equation \( U^* U = I \) holds true, where \( U^* \) is the conjugate transpose and \( I \) is the identity matrix.
  • A unitary matrix is always square, meaning it has the same number of rows and columns.
These qualities make unitary matrices extremely useful in quantum mechanics and signal processing, among other fields.
Unimodularity
Unimodularity refers to the determinant of a matrix being equal to one. A unimodular matrix doesn't alter the volume of any shape in the vector space after transformation, which is crucial in fields like cryptography and mathematical optimization.

For a \(2 \times 2\) matrix, the determinant is calculated as \(ad - bc\) for a matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \). To be unimodular, this value should equal 1. For our specific matrix in the problem:
  • The determinant formula becomes \( a a^* + b^*b \).
  • Given the condition \( a^*a + b^*b = 1 \), the determinant confirms the matrix is unimodular.
By maintaining a determinant of one, unimodular matrices can act as volume-preserving tools which are highly valued in multidimensional geometry and algebraic equations.
Conjugate Transpose
The conjugate transpose, also known as the Hermitian transpose, plays a critical role in the world of matrices. Essentially, it involves taking the transpose of a matrix and then taking the complex conjugate of each entry. This process is particularly important when dealing with complex numbers.

Here鈥檚 how you can find a conjugate transpose:
  • Transpose the matrix. Swap the rows and columns to get \( A^T \).
  • Then take the complex conjugate of each element in \( A^T \) to obtain \( A^* \).
For the matrix in our exercise:
  • The conjugate transpose \( \mathbf{U}^* \) of \( \mathbf{U} = \begin{pmatrix} a & b \ -b^* & a^* \end{pmatrix} \) is \( \begin{pmatrix} a^* & -b \ b^* & a \end{pmatrix} \).
This process is incredibly useful when confirming matrix properties like unitarity, where the relationship \( U^* U = I \) supports the essential identity condition.
Matrix Determinant
The determinant of a matrix is a special number that can be calculated from its elements. For a \(2 \times 2\) matrix, the determinant is usually denoted as \( \, \det(A) \), and calculated using the formula: \[ ad - bc \]where \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \). This value is not just a number, but signifies several important properties of the matrix.

Why determinants matter:
  • Determinants can tell you if a matrix is invertible. If the determinant is not zero, the matrix has an inverse.
  • For a unimodular matrix, this determinant must equal 1, symbolizing a form of consistency and stability in transformations.
  • It provides insights into geometric properties, such as volume scaling factors in transformations.
In the context of this exercise, the determinant supports the unimodular property by confirming as \( a^*a + b^*b = 1 \), helping establish the matrix鈥檚 structural integrity and application significance.

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Most popular questions from this chapter

A plane is covered with reguiar hexagons, as shown in Fig. \(4.17 .\) (a) Determine the dihedral symmetry of an axis perpendicular to the plane through the common vertex of three hexagons ( \(A\) ). That is, if the axis has \(n\) -fold symmetry, show (with careful explanation) what \(n\) is. Write out the \(2 \times 2\) matrix describing the minimum (nonzero) positive rotation of the array of hexagons that is a member of your \(D_{n}\) group. (b) Repeat part (a) for an axis perpendicular to the plane through the geometric center of one hexagon \((B)\).

(a) Show that every four-vector in Minkowski space may be decomposed into an ordinary three-space vector and a three-space scalar. Examples: \((c t, \mathrm{r}),(\rho, \rho \mathrm{v} / c),\left(\varepsilon_{0} \varphi, c \varepsilon_{0} \mathrm{~A}\right),(E / c, \mathrm{p}),(\omega / c, \mathrm{k}) .\) Hint. Consider a rotation of the three-space coordinates with time fixed. (b) Show that the converse of (a) is not true - every three-vector plus scala! does not form a Minkowski four-vector.

The dual of a four-dimensional second-rank tensor \(\mathrm{B}\) may be defined by \(\overline{\mathbf{B}}\). where the elements of the dual tensor are given by $$ \bar{B}^{i j}=\frac{1}{2 !} \varepsilon^{i j k l} B_{k l} $$ Show that \(\overline{\mathrm{B}}\) transforms as (a) a second-rank tensor under rotations, (b) a pseudotensor under inversions.

(i) Show that the Pauli matrices are the generators of \(\mathrm{SU}(2)\) without using the parameterization of the general unitary \(2 \times 2\) matrix in Eq. (4.38). (ii) Derive the eight independent generators \(\lambda_{i}\) of \(\mathrm{SU}(3)\) similarly. Normalize them so that \(\operatorname{tr}\left(\lambda_{i} \lambda_{j}\right)=2 \delta_{i j}\). Then determine the structure constants of \(\operatorname{SU}(3)\). Hint. The \(\lambda_{1}\) are traceless and Hermitian \(3 \times 3\) matrices. (iii) Construct the quadratic Casimir invariant of \(\mathrm{SU}(3)\).

The angular momentum-exponential form of the Euler angle rotation operators is $$ \begin{aligned} R &=R_{z^{\prime \prime}}(\gamma) R_{y^{\prime}}(\beta) R_{z}(\alpha) \\ &=\exp \left(-i \gamma J_{z^{\prime \prime}}\right) \exp \left(-i \beta J_{y^{\prime}}\right) \exp \left(-i \alpha J_{z}\right) . \end{aligned} $$ Show that in terms of the original axes $$ R=\exp \left(i \alpha J_{z}\right) \exp \left(-i \beta J_{y}\right) \exp \left(-i \gamma J_{z}\right) $$ Hint. The \(R\) operators transform as matrices. The rotation about the \(y^{\prime}\) -axi! (second Euler rotation) may be referred to the original \(y\) -axis by $$ \exp \left(-i \beta J_{y}\right)=\exp \left(-i \alpha J_{z}\right) \exp \left(-i \beta J_{y}\right) \exp \left(i \alpha J_{z}\right) . $$
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