Rational canonical form

In linear algebra, the Frobenius normal form or rational canonical form of a square matrix A with entries in a field F is a canonical form for matrices obtained by conjugation by invertible matrices over F. The form reflects a minimal decomposition of the vector space into subspaces that are cyclic for A (i.e., spanned by some vector and its repeated images under A). Since only one normal form can be reached from a given matrix (whence the "canonical"), a matrix B is similar to A if and only if it has the same rational canonical form as A. Since this form can be found without any operations that might change when extending the field F (whence the "rational"), notably without factoring polynomials, this shows that whether two matrices are similar does not change upon field extensions. The form is named after German mathematician Ferdinand Georg Frobenius.

Some authors use the term rational canonical form for a somewhat different form that is more properly called the primary rational canonical form. Instead of decomposing into a minimal number of cyclic subspaces, the primary form decomposes into a maximal number of cyclic subspaces. It is also defined over F, but has somewhat different properties: finding the form requires factorization of polynomials, and as a consequence the primary rational canonical form may change when the same matrix is considered over an extension field of F. This article mainly deals with the form that does not require factorization, and explicitly mentions "primary" when the form using factorization is meant.

When trying to find out whether two square matrices A and B are similar, one approach is to try, for each of them, to decompose the vector space as far as possible a direct sum of stable subspaces, and compare the respective actions on these subspaces. For instance if both are diagonalizable, then one can take the decomposition into eigenspaces (for which the action is as simple as it can get, namely by a scalar), and then similarity can be decided by comparing eigenvalues and their multiplicities. While in practice this is often a quite insightful approach, there are various drawbacks this has as a general method. First, it requires finding all eigenvalues, say as roots of the characteristic polynomial, but it may not be possible to give an explicit expression for them. Second, a complete set of eigenvalues might exist only in an extension of the field one is working over, and then one does not get a proof of similarity over the original field. Finally A and B might not be diagonalizable even over this larger field, in which case one must instead use a decomposition into generalized eigenspaces, and possibly into Jordan blocks.

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