Unitary Matrices#

Definition: Unitary Matrix

A unitary matrix is a complex matrix \(A \in \mathbb{C}^{n\times n}\) which is the matrix representation of some unitary transformation \(T: \mathbb{C}^n \to \mathbb{C}^n\) with respect to the standard basis of \(\mathbb{C}^n\):

\[A = {}_{E_n}[T]_{E_n}\]

Theorem: Length Preservation \(\implies\) Unitary Matrices

A complex square matrix is unitary if and only if multiplying a complex vector \(v \in \mathbb{C}^n\) by it preserves its length:

\[||Av|| = ||v||\]

Proof

TODO

Theorem: Inverses of Unitary Matrices

If \(A \in \mathbb{C}^{n \times n}\) is an unitary matrix, then it is invertible and its inverse is its Hermitian transpose:

\[A^{-1} = A^{\ast}\]

Proof

TODO

Theorem: Determinants of Unitary Matrices

The determinant of an unitary matrix is either \(+1\) or \(-1\).

Proof

TODO

Theorem: Orthonormal Bases from an Unitary Matrix

If \(A\in \mathbb{C}^{n \times n}\) is a unitary matrix, then:

the columns of \(A\) form an orthonormal basis of the complex vector space \(\mathbb{C}^n\);
the rows of \(A\) also form an orthonormal basis of the complex vector space \(\mathbb{C}^n\).

Proof

TODO