A rational matrix is a matrix whose entries are all rational numbers. The set of rational matrices is denoted and is a vector space over the field . After bases are chosen, its elements represent the linear transformations from to .

Rational matrices are closed under matrix addition, multiplication by rational scalars, and matrix multiplication when the dimensions are compatible. Every integer matrix is rational, and every rational matrix is both a real matrix and a complex matrix. Conversely, multiplying a rational matrix by a common denominator of its entries gives an integer matrix.

The Hilbert matrix is a rational matrix whose matrix inverse has entries in the integers. The Pascal matrix and Redheffer matrix are integer, and therefore rational, matrices.

A rational square matrix is invertible over iff its determinant is nonzero. In this case, its matrix inverse is rational by the adjugate matrix formula. Its characteristic polynomial has rational coefficients, although its eigenvalues need not be rational. For example, the rational matrix

has eigenvalues and .

See also

Complex Matrix, Hilbert Matrix, Integer Matrix, Matrix, Pascal Matrix, Rational Canonical Form, Rational Number, Real Matrix, Redheffer Matrix, Square Matrix

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References

Meyer, C. D. Matrix Analysis and Applied Linear Algebra. Philadelphia, PA: SIAM, 2000.

Cite this as:

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

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