Unimodular Matrix
A unimodular matrix is a real square matrix A with determinant det(A)=+/-1 (Born and Wolf 1980, p. 55; Goldstein 1980, p. 149). More generally, a matrix A with elements in the polynomial domain F[x] of a field F is called unimodular if it has an inverse whose elements are also in F[x]. A matrix A is therefore unimodular iff its determinant is a unit of F[x] (MacDuffee 1943, p. 137).
The matrix inverse of a unimodular real matrix is another unimodular matrix.
There are an infinite number of 3×3 unimodular matrices not containing any 0s or +/-1. One parametric family is
![画像: [8n^2+8n 2n+1 4n; 4n^2+4n n+1 2n+1; 4n^2+4n+1 n 2n-1].](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/UnimodularMatrix/NumberedEquation1.svg){kind=link}
Specific examples of unimodular matrices having small positive integer entries include
![画像: [2 3 2; 4 2 3; 9 6 7],[2 3 5; 3 2 3; 9 5 7],[2 3 6; 3 2 3; 17 11 16],...](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/UnimodularMatrix/NumberedEquation2.svg){kind=link}
(Guy 1989, 1994).
The nth power of a unimodular matrix
![画像: M=[m_(11) m_(12); m_(21) m_(22)]](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/UnimodularMatrix/NumberedEquation3.svg){kind=link}
is given by
![画像: M^n=[m_(11)U_(n-1)(a)-U_(n-2)(a) m_(12)U_(n-1)(a); m_(21)U_(n-1)(a) m_(22)U_(n-1)(a)-U_(n-2)(a)],](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/UnimodularMatrix/NumberedEquation4.svg){kind=link}
where
| a=1/2(m_(11)+m_(22)) |
(5)
|
and the U_n are Chebyshev polynomials of the second kind,
![画像: U_m(x)=(sin[(m+1)cos^(-1)x])/(sqrt(1-x^2))](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/UnimodularMatrix/NumberedEquation6.svg){kind=link}
(Born and Wolf 1980, p. 67).
See also
Chebyshev Polynomial of the Second Kind, Determinant, Identity Matrix, Unit MatrixExplore with Wolfram|Alpha
References
Born, M. and Wolf, E. Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. New York: Pergamon Press, pp. 55 and 67, 1980.Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 149, 1980.Guy, R. K. "Unsolved Problems Come of Age." Amer. Math. Monthly 96, 903-909, 1989.Guy, R. K. "A Determinant of Value One." §F28 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 265-266, 1994.MacDuffee, C. C. Vectors and Matrices. Washington, DC: Math. Assoc. Amer., 1943.Séroul, R. Programming for Mathematicians. Berlin: Springer-Verlag, p. 162, 2000.Referenced on Wolfram|Alpha
Unimodular MatrixCite this as:
Weisstein, Eric W. "Unimodular Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UnimodularMatrix.html