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algebraic algorithm angle binary operation binomial coefficients called Cartesian chromatic polynomial coefficients column commutative ring complex numbers components Compute the test converges coordinate system cosh critical value curve defined definition denoted Determine the critical diagonal differential equations edge eigenvalues eigenvectors element example exists F-distribution factor finite finite field formula Fourier given graph G homogeneous coordinates integer integral inverse isomorphic linear m x n matrix Mathematics method modulo multiplication n x n nodes nonzero norm normal distribution orthogonal pair parameter permutations plane polynomial positive integer prime numbers primitive root properties pseudoinverse radius random variable real numbers reject the hypothesis ring rotation sample scalar sequence sin2 sinh solution space square matrix symmetric symmetric matrix tensor theorem transformation triangle vector vertex space vertices zero