Unitary Matrix
Definition
Unitary Matrix
Let $A$ be a square complex matrix. $A$ is called a unitary matrix if it satisfies the following equation:
$$ A^{-1}=A^{\ast} $$
Here, $A^{-1}$ is the inverse of $A$, $A^{\ast}$ is the conjugate transpose of $A$.
Unitary Diagonalization1
Consider a square matrix $A$ of size $n \times n$. $A$ is said to be unitarily diagonalizable if it satisfies the following equation for some diagonal matrix $D$ and unitary matrix $P$:
$$ P^{\ast} A P = D $$
A matrix $P$ that satisfies this condition is said to unitarily diagonalize the matrix $A$.
Description
A unitary matrix is, simply put, an extension of the orthogonal matrix to complex matrices. Therefore, it retains the properties of an orthogonal matrix. The proof of the below equivalence conditions for a unitary matrix is replaced with the proof in Orthogonal Matrix.
Theorem2
Equivalence Conditions for a Unitary Matrix: For a complex matrix $A$ of size $n \times n$, the following propositions are all equivalent.
$A$ is a unitary matrix.
The set of row vectors of $A$ is a normal orthogonal set in $\mathbb{C}^n$.
The set of column vectors of $A$ is a normal orthogonal set in $\mathbb{C}^n$.
$A$ preserves the inner product. That is, for all $\mathbf{x},\mathbf{y}\in \mathbb{C}^{n}$, the following holds:
$$ (A \mathbf{x}) \cdot (A\mathbf{y}) = \mathbf{x} \cdot \mathbf{y} $$
- $A$ preserves length. That is, for all $\mathbf{x}\in \mathbb{C}^{n}$, the following holds:
$$ \left\| A \mathbf{x} \right\| = \left\| \mathbf{x} \right\| $$