Editing Jordan normal form (section)

== Real matrices ==
If ''A'' is a real matrix, its Jordan form can still be non-real. Instead of representing it with complex eigenvalues and ones on the superdiagonal, as discussed above, there exists a real invertible matrix ''P'' such that ''P''<sup>−1</sup>''AP'' = ''J'' is a real [[block diagonal matrix]] with each block being a real Jordan block.<ref>{{harvtxt|Horn|Johnson|1985|loc=Theorem 3.4.5}}</ref> A real Jordan block is either identical to a complex Jordan block (if the corresponding eigenvalue <math>\lambda_i</math> is real), or is a block matrix itself, consisting of 2×2 blocks (for non-real eigenvalue <math>\lambda_i = a_i+ib_i</math> with given algebraic multiplicity) of the form

:<math>C_i = 
\left[ \begin{array}{rr}
a_i  & -b_i \\
b_i & a_i \\ 
\end{array} \right] </math>

and describe multiplication by <math>\lambda_i</math> in the complex plane. The superdiagonal blocks are 2×2 identity matrices and hence in this representation the matrix dimensions are larger than the complex Jordan form. The full real Jordan block is given by

:<math>J_i = 
\begin{bmatrix}
C_i    & I       &        &       \\
       & C_i     & \ddots &       \\     
       &         & \ddots & I     \\
       &         &        & C_i
\end{bmatrix}.</math>

This real Jordan form is a consequence of the complex Jordan form. For a real matrix the nonreal eigenvectors and generalized eigenvectors can always be chosen to form [[complex conjugate]] pairs. Taking the real and imaginary part (linear combination of the vector and its conjugate), the matrix has this form with respect to the new basis.