Spin and Pin
A subtle technical problem afflicts some uses of orthogonal matrices. Not only are the group components with determinant +1 and −1 not connected to each other, even the +1 component, SO(n), is not simply connected (except for SO(1), which is trivial). Thus it is sometimes advantageous, or even necessary, to work with a covering group of SO(n), the spin group, Spin(n). Likewise, O(n) has covering groups, the pin groups, Pin(n). For n > 2, Spin(n) is simply connected, and thus the universal covering group for SO(n). By far the most famous example of a spin group is Spin(3), which is nothing but SU(2), or the group of unit quaternions.
The Pin and Spin groups are found within Clifford algebras, which themselves can be built from orthogonal matrices.
Read more about this topic: Orthogonal Matrix
Famous quotes containing the words spin and, spin and/or pin:
“Spin and die,
To live again as butterfly.”
—Christina Georgina Rossetti (1830–1894)
“In tragic life, God wot,
No villain need be! Passions spin the plot:
We are betrayed by what is false within.”
—George Meredith (1828–1909)
“A man is a beggar who only lives to the useful, and, however he may serve as a pin or rivet in the social machine, cannot be said to have arrived at self-possession.”
—Ralph Waldo Emerson (1803–1882)