Editing Frobenius normal form (section)

== Example ==
Consider the following matrix A, over '''Q''':

:<math>\scriptstyle A=\begin{pmatrix}
     -1& 3&-1& 0&-2& 0& 0&-2 \\
     -1&-1& 1& 1&-2&-1& 0&-1 \\
     -2&-6& 4& 3&-8&-4&-2& 1 \\
     -1& 8&-3&-1& 5& 2& 3&-3 \\
      0& 0& 0& 0& 0& 0& 0& 1 \\
      0& 0& 0& 0&-1& 0& 0& 0 \\
      1& 0& 0& 0& 2& 0& 0& 0 \\
      0& 0& 0& 0& 4& 0& 1& 0 \end{pmatrix}.</math>

''A'' has [[minimal polynomial (linear algebra)|minimal polynomial]] <math>\mu = X^6 - 4X^4 - 2X^3 + 4X^2 + 4X + 1</math>, so that the [[dimension (vector space)|dimension]] of a subspace generated by the repeated images of a single vector is at most 6. The [[characteristic polynomial]] is <math>\chi = X^8 - X^7 - 5X^6 + 2X^5 + 10X^4 + 2X^3 - 7X^2 - 5X - 1</math>, which is a multiple of the minimal polynomial by a factor <math>X^2 - X - 1</math>. There always exist vectors such that the cyclic subspace that they generate has the same minimal polynomial as the operator has on the whole space; indeed most vectors will have this property, and in this case the first standard basis vector <math>e_1</math> does so: the vectors <math>A^k(e_1)</math> for <math>k=0,1,\ldots,5</math> are linearly independent and span a cyclic subspace with minimal polynomial <math>\mu</math>. There exist complementary stable subspaces (of dimension 2) to this cyclic subspace, and the space generated by vectors <math>v=(3,4,8,0,-1,0,2,-1)^\top</math> and <math>w=(5,4,5,9,-1,1,1,-2)^\top</math> is an example. In fact one has <math>A\cdot v=w</math>, so the complementary subspace is a cyclic subspace generated by <math>v</math>; it has minimal polynomial <math>X^2 - X - 1</math>. Since <math>\mu</math> is the minimal polynomial of the whole space, it is clear that <math>X^2 - X - 1</math> must divide <math>\mu</math> (and it is easily checked that it does), and we have found the invariant factors <math>X^2 - X - 1</math> and <math>\mu = X^6 - 4X^4 - 2X^3 + 4X^2 + 4X + 1</math> of ''A''. Then the rational canonical form of ''A'' is the [[block diagonal matrix]] with the corresponding companion matrices as diagonal blocks, namely
:<math>\scriptstyle C=\left(\begin{array}{cc|cccccc}
      0& 1& 0& 0& 0& 0& 0& 0 \\
      1& 1& 0& 0& 0& 0& 0& 0 \\\hline
      0& 0& 0& 0& 0& 0& 0&-1 \\
      0& 0& 1& 0& 0& 0& 0&-4 \\
      0& 0& 0& 1& 0& 0& 0&-4 \\
      0& 0& 0& 0& 1& 0& 0& 2 \\
      0& 0& 0& 0& 0& 1& 0& 4 \\
      0& 0& 0& 0& 0& 0& 1& 0 \end{array}\right).</math>
A basis on which this form is attained is formed by the vectors <math>v,w</math> above, followed by <math>A^k(e_1)</math> for <math>k=0,1,\ldots,5</math>; explicitly this means that for
:<math>\scriptstyle P=\begin{pmatrix}
      3& 5& 1&-1& 0& 0& -4& 0\\
      4& 4& 0&-1&-1&-2& -3&-5\\
      8& 5& 0&-2&-5&-2&-11&-6\\
      0& 9& 0&-1& 3&-2&  0& 0\\
     -1&-1& 0& 0& 0& 1& -1& 4\\
      0& 1& 0& 0& 0& 0& -1& 1\\
      2& 1& 0& 1&-1& 0&  2&-6\\
     -1&-2& 0& 0& 1&-1&  4&-2 \end{pmatrix}</math>,

one has <math>A=PCP^{-1}.</math>