A square matrix A is orthogonal if A^TA = I, where A^T is the transpose of A. The following properties hold:

1. (i) If A is orthogonal, A⁻¹ = A^T and so AA^T = I.

2. (ii) If A is orthogonal, det A = ±1.

3. (iii) If A and B are orthogonal matrices of the same order, then AB is orthogonal, as is A⁻¹.

The orthogonal n × n matrices form a group O(n), and those with determinant 1 form a subgroup SO(n). The orthogonal matrices are the linear (see linear map) isometries of ℝⁿ.

2 × 2 orthogonal matrices have the form

The first matrix represents rotation by θ‎ anticlockwise about the origin. The second represents reflection in the line y = xtan(θ‎/2). See unitary matrix.