A unimodular matrix is a real square matrix determinant field iff its determinant
is a unit of
The matrix inverse of a unimodular real
matrix is another unimodular matrix.
There are an infinite number of
(1)
Specific examples of unimodular matrices having small positive integer entries include
(2)
(Guy 1989, 1994).
The power of a unimodular matrix
(3)
is given by
(4)
where
(5)
and the Chebyshev polynomials of the
second kind ,
(6)
(Born and Wolf 1980, p. 67).
See also Chebyshev Polynomial of the Second Kind ,
Determinant ,
Identity
Matrix ,
Unit Matrix
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References Born, M. and Wolf, E. Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of
Light, 6th ed. Goldstein,
H. Classical
Mechanics, 2nd ed. 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. MacDuffee, C. C. Vectors
and Matrices. Séroul,
R. Programming
for Mathematicians. Referenced
on Wolfram|Alpha Unimodular Matrix
Cite this as:
Weisstein, Eric W. "Unimodular Matrix."
From MathWorld https://mathworld.wolfram.com/UnimodularMatrix.html
Subject classifications
with determinant 
(Born and Wolf 1980, p. 55; Goldstein 1980,
p. 149). More generally, a matrix 
with elements in the polynomial domain ![F[x]](/images/equations/UnimodularMatrix/Inline4.svg)
of a field 
is called unimodular if it has an inverse whose elements are
also in ![F[x]](/images/equations/UnimodularMatrix/Inline6.svg)
.
A matrix 
is therefore unimodular iff its determinant
is a unit of ![F[x]](/images/equations/UnimodularMatrix/Inline8.svg)
(MacDuffee 1943, p. 137).
unimodular matrices not containing any 0s or 
. One parametric family is
![[8n^2+8n 2n+1 4n; 4n^2+4n n+1 2n+1; 4n^2+4n+1 n 2n-1].](/images/equations/UnimodularMatrix/NumberedEquation1.svg)
![[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],...](/images/equations/UnimodularMatrix/NumberedEquation2.svg)
th
power of a unimodular matrix
![M=[m_(11) m_(12); m_(21) m_(22)]](/images/equations/UnimodularMatrix/NumberedEquation3.svg)
![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)],](/images/equations/UnimodularMatrix/NumberedEquation4.svg)
are Chebyshev polynomials of the
second kind,
![U_m(x)=(sin[(m+1)cos^(-1)x])/(sqrt(1-x^2))](/images/equations/UnimodularMatrix/NumberedEquation6.svg)
