Unitary Matrix

• categories: Linear algebra, Matrix

Definition

A square matrix $U\in {\mathbb{C}}^{n\times n}$ is called a unitary matrix if its conjugate transpose equals its inverse:

$$U^{\ast }U=U{U}^{\ast }=I,$$

where:

• $U^{\ast }$ is the conjugate transpose of $U$ (i.e., ${\overline{U}}^T$, where $\overline{U}$ is the element-wise complex conjugate of $U$),
• $I$ is the $n\times n$ identity matrix.

For real matrices, a unitary matrix is equivalent to an Orthogonal matrix

Key Properties

1. Preservation of Norms:
   For any vector $\mathbf{x}\in {\mathbb{C}}^n$:

   $$\Vert U\mathbf{x}{\Vert }_2=\Vert \mathbf{x}{\Vert }_2.$$

2. Eigenvalues:
   The eigenvalues of a unitary matrix lie on the complex unit circle, i.e., $|\lambda |=1$.

3. Determinant:
   The determinant of a unitary matrix has absolute value 1:

   $$|\det (U)|=1.$$

4. Preservation of Inner Products:
   For any vectors $\mathbf{x},\mathbf{y}\in {\mathbb{C}}^n$:

   $$⟨U\mathbf{x},U\mathbf{y}⟩=⟨\mathbf{x},\mathbf{y}⟩,$$

   where $⟨\mathbf{x},\mathbf{y}⟩={\mathbf{x}}^{\ast }\mathbf{y}$ is the standard inner product.

5. Inverse:
   The inverse of a unitary matrix is its conjugate transpose:

   $$U^{-1}=U^{\ast }.$$

6. Stability Under Multiplication:
   The product of two unitary matrices is also unitary:

   $$U_1{U}_2\text{ is unitary if }{U}_1\text{ and }{U}_2\text{ are unitary.}$$