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Frobenius normal form
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<p>In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, the Frobenius normal form or rational canonical form of a <a href="/facts/Square_matrix/5TP9mNNn">square</a> <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> <i>A</i> with entries in a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> <i>F</i> is a <a href="/facts/Canonical_form/K2jhB44k">canonical form</a> for matrices obtained by conjugation by <a href="/facts/Invertible_matrices/fPqXk3V8">invertible matrices</a> over <i>F</i>. The form reflects a minimal decomposition of the <a href="/facts/Vector_space/M1pxMLx2">vector space</a> into <a href="/facts/Linear_subspace/2wtKTgHL">subspaces</a> that are cyclic for <i>A</i> (i.e., <a href="/facts/Linear_span/vPjclEtb">spanned</a> by some vector and its repeated <a href="/facts/Image_(mathematics)/KANefMAl">images</a> under <i>A</i>). Since only one normal form can be reached from a given matrix (whence the "canonical"), a matrix <i>B</i> is <a href="/facts/Matrix_similarity/ySevpav1">similar</a> to <i>A</i> if and only if it has the same rational canonical form as <i>A</i>. Since this form can be found without any operations that might change when <a href="/facts/Field_extension/L7yIDtyK">extending</a> the field <i>F</i> (whence the "rational"), notably without <a href="/facts/Factorization_of_polynomials/1S9HRKcI">factoring polynomials</a>, this shows that whether two matrices are similar does not change upon field extensions. The form is named after German mathematician <a href="/facts/Ferdinand_Georg_Frobenius/4cXXQyu5">Ferdinand Georg Frobenius</a>. </p><p>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 minimum number of cyclic subspaces, the primary form decomposes into a maximum number of cyclic subspaces. It is also defined over <i>F</i>, 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 <i>F</i>. This article mainly deals with the form that does not require factorization, and explicitly mentions "primary" when the form using factorization is meant. </p>