Unitary matrices: A complex, square matrix U is called...

Unitary matrices: A complex, square matrix $U$ is called unitary if it satisfies $U^{\dagger} U=\mathrm{I}$, where $U^{\dagger}=\overline{U^T}$ denotes the Hermitian adjoint in which one first transposes and then takes complex conjugates of all entries. (a) Show that $U$ is a unitary matrix if and only if $U^{-1}=U^{\dagger}$. (b) Show that the following matrices are unitary and compute their inverses: (i) $\left(\begin{array}{ll}\frac{1}{\sqrt{2}} & \frac{\mathrm{i}}{\sqrt{2}} \\ \frac{\mathrm{i}}{\sqrt{2}} & \frac{1}{\sqrt{2}}\end{array}\right)$, (ii) $\left(\begin{array}{ccc}\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} & -\frac{1}{2 \sqrt{3}}+\frac{1}{2} & -\frac{1}{2 \sqrt{3}}-\frac{1}{2} \\ \frac{1}{\sqrt{3}} & -\frac{1}{2 \sqrt{3}}-\frac{\mathrm{i}}{2} & -\frac{1}{2 \sqrt{3}}+\frac{\mathrm{i}}{2}\end{array}\right)$, (iii) $$ \left(\begin{array}{rrrr} \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{i}{2} & -\frac{1}{2} & -\frac{i}{2} \\ \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \\ \frac{1}{2} & -\frac{i}{2} & -\frac{1}{2} & \frac{i}{2} \end{array}\right) $$ (c) Are the following matrices unitary? (i) $\left(\begin{array}{cc}2 & 1+2 \mathrm{i} \\ 1-2 \mathrm{i} & 3\end{array}\right)$, (ii) $\frac{1}{5}\left(\begin{array}{cc}-1+2 \mathrm{i} & -4-2 \mathrm{i} \\ 2-4 \mathrm{i} & -2-\mathrm{i}\end{array}\right)$, (iii) $\left(\begin{array}{rr}\frac{12}{13} & \frac{5}{13} \\ \frac{5}{13} & -\frac{12}{13}\end{array}\right)$. (d) Show that $U$ is a unitary matrix if and only if its columns form an orthonormal basis of $\mathbb{C}^n$ with respect to the Hermitian dot product. (e) Prove that the set of unitary matrices forms a group, as defined in Exercise 4.3.24.

Unitary matrices: A complex, square matrix $U$ is called unitary if it satisfies $U^{\dagger} U=\mathrm{I}$, where $U^{\dagger}=\overline{U^T}$ denotes the Hermitian adjoint in which one first transposes and then takes complex conjugates of all entries. (a) Show that $U$ is a unitary matrix if and only if $U^{-1}=U^{\dagger}$. (b) Show that the following matrices are unitary and compute their inverses:
(i) $\left(\begin{array}{ll}\frac{1}{\sqrt{2}} & \frac{\mathrm{i}}{\sqrt{2}} \\ \frac{\mathrm{i}}{\sqrt{2}} & \frac{1}{\sqrt{2}}\end{array}\right)$,
(ii) $\left(\begin{array}{ccc}\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} & -\frac{1}{2 \sqrt{3}}+\frac{1}{2} & -\frac{1}{2 \sqrt{3}}-\frac{1}{2} \\ \frac{1}{\sqrt{3}} & -\frac{1}{2 \sqrt{3}}-\frac{\mathrm{i}}{2} & -\frac{1}{2 \sqrt{3}}+\frac{\mathrm{i}}{2}\end{array}\right)$,
(iii)
$$
\left(\begin{array}{rrrr}
\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & \frac{i}{2} & -\frac{1}{2} & -\frac{i}{2} \\
\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \\
\frac{1}{2} & -\frac{i}{2} & -\frac{1}{2} & \frac{i}{2}
\end{array}\right)
$$
(c) Are the following matrices unitary?
(i) $\left(\begin{array}{cc}2 & 1+2 \mathrm{i} \\ 1-2 \mathrm{i} & 3\end{array}\right)$,
(ii) $\frac{1}{5}\left(\begin{array}{cc}-1+2 \mathrm{i} & -4-2 \mathrm{i} \\ 2-4 \mathrm{i} & -2-\mathrm{i}\end{array}\right)$,
(iii) $\left(\begin{array}{rr}\frac{12}{13} & \frac{5}{13} \\ \frac{5}{13} & -\frac{12}{13}\end{array}\right)$.
(d) Show that $U$ is a unitary matrix if and only if its columns form an orthonormal basis of $\mathbb{C}^n$ with respect to the Hermitian dot product. (e) Prove that the set of unitary matrices forms a group, as defined in Exercise 4.3.24.

Key Concepts

Group Structure of Unitary Matrices

The set of all unitary matrices of a fixed size forms a group under matrix multiplication. This means that the product of two unitary matrices is unitary, there exists an identity unitary matrix, each unitary matrix has an inverse (given by its Hermitian adjoint), and the group operation (matrix multiplication) is associative. This group, often denoted U(n) for n×n matrices, is fundamental in both mathematics and physics for studying symmetries and conservation laws.

Matrix Inverse and Its Relation to the Adjoint

For a unitary matrix, an important characteristic is that its inverse is equal to its Hermitian adjoint, i.e., U?1 = U†. This equivalence not only simplifies the calculation of inverses for such matrices but also underscores the inherent symmetry and norm-preservation properties of unitary matrices.

Orthonormal Basis in Complex Vector Spaces

In the context of complex vector spaces, the columns of a unitary matrix form an orthonormal basis with respect to the Hermitian (or complex) dot product. This means that the inner product of any two distinct columns is zero and each column has unit norm. This property is essential for many applications, such as simplifying the analysis of linear transformations and quantum state representations.

Unitary Matrix

A unitary matrix is a complex square matrix U that satisfies U†U = I, where U† is the Hermitian adjoint of U. This property implies that unitary matrices preserve the length of vectors and are norm-preserving isometries in complex vector spaces. In other words, they maintain the inner product structure, analogous to orthogonal matrices in real vector spaces.

Hermitian Adjoint

The Hermitian adjoint (or conjugate transpose) of a matrix U, denoted by U†, is obtained by first transposing U and then taking the complex conjugate of each entry. This operation is critical in the definition of unitary matrices and plays a central role in many areas of linear algebra and quantum mechanics, as it relates to the preservation of inner products.

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