- Author:
- Mark William Giesbrecht
Computing canonical forms of matrices is a classical mathematical problem with many practical applications. We investigate the problem of computing the Frobenius canonical form and the more familiar Jordan canonical form of a matrix over any field. A probabilistic algorithm is exhibited which computes the Frobenius form of a matrix in about the same number of operations asymptotically as is required for matrix multiplication. It is also demonstrated that determining the Frobenius form is at least as hard as matrix multiplication, whence our algorithm is nearly optimal. Finding the Jordan form of a matrix is reduced to finding the Frobenius form and factoring the characteristic polynomial of that matrix.
We adapt our algorithms for computing the Frobenius and Jordan forms of matrices over general fields to practical algorithms for real and rational matrices. Canonical forms such as these, which capture all the geometric information in a matrix, are numerically highly unstable, and lend themselves to exact (as opposed to floating point) arithmetic. We take this approach and demonstrate efficient probabilistic algorithms to find these canonical forms exactly.
Applications of our techniques to important problems such as matrix powering, evaluating a polynomial at a matrix, determining matrix similarity and finding special purpose bases for finite fields are also explored. Many of these algorithms also match corresponding lower bounds for these problems which we also demonstrate.
Finally, we consider the more complex problems of decomposing finite associative algebras and factoring in non-commutative polynomial rings. We use our probabilistic algorithms for computing Frobenius forms of matrices, as well as new lower bounds on the density of elements with certain properties in any finite associative algebra, to obtain fast new algorithms for these difficult problems.
Cited By
Bostan A, Caruso X and Schost É Computation of the Similarity Class of the p-Curvature Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, (111-118)- Pernet C and Storjohann A Faster algorithms for the characteristic polynomial Proceedings of the 2007 international symposium on Symbolic and algebraic computation, (307-314)
- Giorgi P, Jeannerod C and Villard G On the complexity of polynomial matrix computations Proceedings of the 2003 international symposium on Symbolic and algebraic computation, (135-142)
- Fortuna E and Gianni P (1999). Square-free decomposition in finite characteristic, ACM SIGSAM Bulletin, 33:4, (14-32), Online publication date: 1-Dec-1999.
- Storjohann A An O(n3) algorithm for the Frobenius normal form Proceedings of the 1998 international symposium on Symbolic and algebraic computation, (101-105)
- Giesbrecht M Fast computation of the Smith normal form of an integer matrix Proceedings of the 1995 international symposium on Symbolic and algebraic computation, (110-118)
- Kaltofen E and Shoup V Subquadratic-time factoring of polynomials over finite fields Proceedings of the twenty-seventh annual ACM symposium on Theory of computing, (398-406)
- Giesbrecht M Fast algorithms for rational forms of integer matrices Proceedings of the international symposium on Symbolic and algebraic computation, (305-311)
- Villard G Fast parallel computation of the Smith normal form of polynomial matrices Proceedings of the international symposium on Symbolic and algebraic computation, (312-317)
Index Terms
- Nearly optimal algorithms for canonical matrix forms
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