Orthogonal Matrix

Definition

Let $A$ be a square real matrix. $A$ is called an orthogonal matrix if it satisfies the following equation:

$$ A^{-1} = A^{T} $$

Another way to express this condition is as follows:

$$ AA^{T} = A^{T}A =I $$

Explanation

To put the definition in words, an orthogonal matrix is a matrix whose row vectors or column vectors are orthogonal unit vectors to each other. When extended to complex matrices, it is called a unitary matrix. A concrete example of an orthogonal matrix is the rotation matrix. The transformation that rotates a vector in the 2D plane counterclockwise by $\theta$ is as follows:

$$ A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} $$

From the following formula, it can be seen that the rotation transformation is an orthogonal matrix for any $\theta$.

$$ A^{T} A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I $$

Properties

The transpose of an orthogonal matrix is an orthogonal matrix.
The inverse of an orthogonal matrix is an orthogonal matrix.
The product of two orthogonal matrices is an orthogonal matrix.
The determinant of an orthogonal matrix is either $1$ or $-1$.

$$ \det(A)=\pm 1 $$

Equivalent Conditions for an Orthogonal Matrix

For a real matrix $A$, the following propositions are all equivalent:

$A$ is an orthogonal matrix.
The set of row vectors of $A$ forms a(n) $\mathbb{R}^n$ orthonormal set.
The set of column vectors of $A$ forms a(n) $\mathbb{R}^n$ orthonormal set.
$A$ preserves inner product. That is, for all $\mathbf{x},\mathbf{y}\in \mathbb{R}^{n}$, the following holds:

$$ (A \mathbf{x}) \cdot (A\mathbf{y}) = \mathbf{x} \cdot \mathbf{y} $$

$A$ preserves length. That is, for all $\mathbf{x}\in \mathbb{R}^{n}$, the following holds:

$$ \left\| A \mathbf{x} \right\| = \left\| \mathbf{x} \right\| $$