orthogonal matrix
An orthogonal matrix is a square matrix whose columns form an orthonormal set: each column has length 1, and any two different columns are perpendicular. The columns are a ready-made orthonormal basis.
What it does geometrically is a rigid motion, a rotation or a reflection, that preserves every length and every angle. Apply it to a shape and the shape turns or flips but never stretches, shrinks, or skews. Distances between points stay exactly the same.
This makes its inverse remarkably cheap. Instead of the usual heavy inversion, you just flip it across its diagonal: Q^T = Q^(-1). That single fact, transpose equals inverse, is the defining shortcut of orthogonal matrices.
Transpose times matrix gives the identity, so Q^T is the inverse.
Its determinant is always +1 (a rotation) or -1 (a reflection), reflecting that it never changes volume.