Orthogonal Matrices

Linear Algebra (Matrices) · Axiom Academy

A square matrix whose columns are orthonormal — the rigid rotations and reflections that preserve every length and angle. 1. Columns That Form a Perfect Frame Build Q one column at a time. Watch each column grow to exactly unit length and lock in perpendicular to the other. When the columns are orthonormal, Q is orthogonal . The product collects every pairwise dot product of the columns. Orthonormal columns make the off-diagonal dot products 0 and the diagonal squared norms 1 — which is precisely the identity matrix. 2. A Rigid Motion: Lengths and Angles Survive Apply to the unit square (left). It rotates rigidly — every side stays length 1 and every corner stays a right angle. A non-orthogonal matrix (right) with the same determinant 1 shears the square: areas match, but lengths and angles are wrecked. For every vector : . The rotation cannot stretch or shrink. For every : . Perpendicular stays perpendicular. The shear also has , yet — so it distorts. Every entry of is a dot product of two columns of Q . Watch each cell compute: the off-diagonals are (the columns are perpendicular) and the diagonal are (the columns are unit). The grid resolves to the identity. An orthogonal matrix is a rigid motion in disguise: orthonormal columns, , and every length and angle untouched. Scroll up to revisit any step.

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