Linear algebra/Orthogonal matrix

Alternative notations

A real-valued and square matrix is orthogonal (or orthonormal) if where is the transpose and is its inverse. Equivalently, is orthogonal when where is the identity matrix. It has a number of well-known properties:[1]


Since orthogonal matrices form a group under multiplication, we can construct a non-trivial orthogonal matrix my multiplying two matrices that are easy to understand. Consider, for example,

Both and posses a symmetry that could lead one to postulate a non-existent symmetry among off-diagonal elements. But no such symmetry exists for . The upper-left 2x2 submatrix in represent a 45 degree rotation around the (or ) axis, plus and inversion through that same axis, i.e., .