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A unimodular matrix is a real square matrix with determinant (Born and Wolf 1980, p. 55; Goldstein 1980, p. 149). More generally, a matrix with elements in the polynomial domain of a field is called unimodular if it has an inverse whose elements are also in . A matrix is therefore unimodular iff its determinant is a unit of (MacDuffee 1943, p. 137).

The matrix inverse of a unimodular real matrix is another unimodular matrix.

There are an infinite number of unimodular matrices not containing any 0s or . One parametric family is

Specific examples of unimodular matrices having small positive integer entries include

(Guy 1989, 1994).

The th power of a unimodular matrix

is given by

where

and the are Chebyshev polynomials of the second kind,

(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. 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, Germany: Springer-Verlag, p. 162, 2000.

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Unimodular Matrix

Cite this as:

Weisstein, Eric W. "Unimodular Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UnimodularMatrix.html

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