Unimodular Matrix

A Matrix ${\hbox{\sf A}}$ with Integer elements and Determinant det $({\hbox{\sf A}})=$ ± 1, also called a Unit Matrix.

The inverse of a unimodular matrix is another unimodular matrix. A Positive unimodular matrix has det $({\hbox{\sf A}})=+1$. The $n$th Power of a Positive Unimodular Matrix

$${\hbox{\sf M}}\equiv \left[{\matrix{m_{11} & m_{12}\cr m_{21} & m_{22}\cr}}\right]$$ (1)

is

$${\hbox{\sf M}}^n = \left[{\matrix{ m_{11}U_{n-1}(a)-U_{n-2}(... ...cr m_{21}U_{n-1}(a) & m_{22}U_{n-1}(a)-U_{n-2}(a)\cr}}\right],$$ (2)

where

$$a\equiv {\textstyle{1\over 2}}(m_{11}+m_{22})$$ (3)

and the $U_n$ are Chebyshev Polynomials of the Second Kind,

$$U_m(x)={\sin[(m+1)\cos^{-1} x]\over\sqrt{1-x^2}}.$$ (4)

See also Chebyshev Polynomial of the Second Kind

References

Born, M. and Wolf, E. Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. New York: Pergamon Press, p. 67, 1980.

Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 149, 1980.