An orthonormal matrix has the property that its transpose and its inverse matrix are equal, or, equivalently, that any pair from the set of column vectors (or row vectors) of the matrix are "perpendicular" or orthogonal (have a scalar product of zero), while any individual vector from the set has a norm (scalar product with itself) of one. Briefly, the set of row or column vector forms an orthonormal set. It can thus be considered as a transformation between one mutually perpendicular set of axes and another, forming Cartesian coordinate systems. The axes of one coordinate system may be read off in terms of the other as the row vectors or column vectors of the matrix. In three dimensions, an orthonormal matrix represents an arbitrary rotation about the origin, or reflection in some plane through the origin. It is almost immediate from the definition that the inverse of an orthonormal matrix is itself orthonormal.

Of essential importance in 3D computer graphics, since an orthonormal matrix can represent the rotation of any object about its origin, including the camera. In three dimensions, the nine matrix elements are inherently dependent and have only three degrees of freedom, analogous to pitch, roll, and yaw. Various representations of an orthonormal matrix exist using only three parameters, some allowing the transformation to be performed, or multiple transformations to be concatenated more efficiently than by matrix multiplication in Cartesian coordinates.

Log in or register to write something here or to contact authors.