Unitary and orthogonal matrices

Orthogonal matrix

Definition

A real square matrix \(U\) is called orthogonal if the columns of \(U\) form an orthonormal set. In other words, let

\[U = \begin{bmatrix} u_1 & u_2 & \dots & u_n \end{bmatrix}\]

with \(u_i \in \RR^n\). Then we have

\[u_i \cdot u_j = \delta_{i , j}.\]

Lemma

An orthogonal matrix \(U\) is invertible with \(U^T = U^{-1}\).

Proof

Let

\[U = \begin{bmatrix} u_1 & u_2 & \dots & u_n \end{bmatrix}\]

be orthogonal with

\[\begin{split}U^T = \begin{bmatrix} u_1^T \\ u_2^T \\ \vdots \\ u_n^T. \end{bmatrix}\end{split}\]

Then

\[\begin{split}U^T U = \begin{bmatrix} u_1^T \\ u_2^T \\ \vdots \\ u_n^T. \end{bmatrix} \begin{bmatrix} u_1 & u_2 & \dots & u_n \end{bmatrix} = \begin{bmatrix} u_i \cdot u_j \end{bmatrix} = I.\end{split}\]

Since columns of \(U\) are linearly independent and span \(\RR^n\), hence \(U\) is invertible. Thus

\[U^T = U^{-1}.\]

Lemma

Determinant of an orthogonal matrix is \(\pm 1\).

Proof

Let \(U\) be an orthogonal matrix. Then

\[\det (U^T U) = \det (I) \implies \left ( \det (U) \right )^2 = 1\]

Thus we have

\[\det(U) = \pm 1.\]

Unitary matrix

Definition

A complex square matrix \(U\) is called unitary if the columns of \(U\) form an orthonormal set. In other words, let

\[U = \begin{bmatrix} u_1 & u_2 & \dots & u_n \end{bmatrix}\]

with \(u_i \in \CC^n\). Then we have

\[u_i \cdot u_j = \langle u_i , u_j \rangle = u_j^H u_i = \delta_{i , j}.\]

Lemma

A unitary matrix \(U\) is invertible with \(U^H = U^{-1}\).

Proof

Let

\[U = \begin{bmatrix} u_1 & u_2 & \dots & u_n \end{bmatrix}\]

be orthogonal with

\[\begin{split}U^H = \begin{bmatrix} u_1^H \\ u_2^H \\ \vdots \\ u_n^H. \end{bmatrix}\end{split}\]

Then

\[\begin{split}U^H U = \begin{bmatrix} u_1^H \\ u_2^H \\ \vdots \\ u_n^H. \end{bmatrix} \begin{bmatrix} u_1 & u_2 & \dots & u_n \end{bmatrix} = \begin{bmatrix} u_i^H u_j \end{bmatrix} = I.\end{split}\]

Since columns of \(U\) are linearly independent and span \(\CC^n\), hence \(U\) is invertible. Thus

\[U^H = U^{-1}.\]

Lemma

The magnitude of determinant of a unitary matrix is \(1\).

Proof

Let \(U\) be a unitary matrix. Then

\[\det (U^H U) = \det (I) \implies \det(U^H) \det(U) = 1 \implies \overline{\det(U)}{\det(U)} = 1.\]

Thus we have

\[|\det(U) |^2 = 1 \implies |\det(U) | = 1.\]

F unitary matrix

We provide a common definition for unitary matrices over any field \(\FF\). This definition applies to both real and complex matrices.

Definition

A square matrix \(U \in \FF^{n \times n}\) is called \(\FF\)-unitary if the columns of \(U\) form an orthonormal set. In other words, let

\[U = \begin{bmatrix} u_1 & u_2 & \dots & u_n \end{bmatrix}\]

with \(u_i \in \FF^n\). Then we have

\[\langle u_i , u_j \rangle = u_j^H u_i = \delta_{i , j}.\]

We note that a suitable definition of inner product transports the definition appropriately into orthogonal matrices over \(\RR\) and unitary matrices over \(\CC\).

When we are talking about \(\FF\) unitary matrices, then we will use the symbol \(U^H\) to mean its inverse. In the complex case, it will map to its conjugate transpose, while in real case it will map to simple transpose.

This definition helps us simplify some of the discussions in the sequel (like singular value decomposition).

Following results apply equally to orthogonal matrices for real case and unitary matrices for complex case.

Lemma

\(\FF\)-unitary matrices preserve norm. i.e.

\[\| U x \|_2 = \|x \|_2.\]

Proof

\[\| U x \|_2^2 = (U x)^H (U x) = x^H U^H U x = x^H I x = \| x\|_2^2.\]

Remark

For the real case we have

\[\| U x \|_2^2 = (U x)^T (U x) = x^T U^T U x = x^T I x = \| x\|_2^2.\]

Lemma

\(\FF\)-unitary matrices preserve inner product. i.e.

\[\langle U x, U y \rangle = \langle x, y \rangle.\]

Proof

\[\langle U x, U y \rangle = (U y)^H U x = y^H U^H U x = y^H x.\]

Remark

For the real case we have

\[\langle U x, U y \rangle = (U y)^T U x = y^T U^T U x = y^T x.\]