Orthogonal matrix

Definition

Let $n \ge 1$ and let $Q$ be a real $n \times n$ matrix. The matrix $Q$ is orthogonal when $$Q^T Q = I_n,$$ where $Q^T$ is the transpose and $I_n$ is the identity matrix.

Remarks

Equivalently, $Q$ is invertible with $Q^{-1} = Q^T$, and then also $Q Q^T = I_n$. Equivalently, the columns of $Q$ form an orthonormal basis of $\R^n$; the same holds for the rows. The map $x \mapsto Qx$ keeps the dot product, so it is a linear isometry of $\R^n$. The determinant of an orthogonal matrix is $1$ or $-1$.

Explanations

L1 human

Orthogonal matrices describe the rigid motions of space that fix the origin: rotations and reflections. They keep all lengths and angles. To invert an orthogonal matrix, we only need to transpose it.