### Invariant subspaces of linear maps: Invariant subspaces

### From real to complex vector spaces and back

Before we will deal with normal forms for real matrices having a characteristic polynomial with non-real roots, we will discuss the extension of a real vector space towards a complex one and the question how the original real vector space can be found in the complex one.

Let #V# be a real vector space. Take a look at the set

\[V_{\mathbb{C} }= V + \ii V\]

consisting of all vectors of the form #\vec{x}+\ii\vec{y}#, where #\vec{x}# and #\vec{y}# are vectors of #V#, and the addition is formal (hence, can also be seen as the pair #\rv{\vec{x},\vec{y}}#). This is a complex vector space if it is provided with the following operations:

\[\begin{array}{rclcl}\text{scalar multiplication}&:&(a+b\ii)(\vec{x}+\ii\vec{y}) &=&(a\cdot\vec{x}-b\cdot\vec{y})+\ii(b\cdot\vec{x}+a\cdot\vec{y})\\ \text{vector addition}&:&(\vec{x}+\ii\vec{y}) +(\vec{u}+\ii\vec{v})&=&(\vec{x}+\vec{u})+\ii(\vec{y}+\vec{v})\end{array}\] We will call this the **extension** of #V# **to a complex vector space**.

The map #\sigma : V_{\mathbb{C} }\to V_{\mathbb{C} }# defined by #\sigma(\vec{x}+\ii\vec{y}) = \vec{x}-\ii\vec{y}# for \(\vec{x},\vec{y}\) in #V# is called **complex conjugate**. This map is **semi-linear**, which means that, for all complex numbers #\lambda# and #\mu# and all vectors #\vec{u}# and #\vec{w}# of #V_{\mathbb{C} }# we have

\[\sigma(\lambda\cdot\vec{u}+\mu\cdot\vec{w}) = \overline{\lambda}\cdot\sigma(\vec{u}) +\overline{\mu}\cdot\sigma(\vec{w})\] where #\overline{\lambda}# indicates the complex conjugate of #\lambda#.

To view a complex vector space as a real space, no expansion is needed:

A complex vector space #W# can be seen as a real vector space by **limitation of scalars**. This means we will only view the real numbers of #W# as scalars. We will write #\left.W\right|_{\mathbb{R}}# to indicate this real vector space.

Let #V# be a real vector space. After limitation of scalars for #W = V_{\mathbb{C} }# the semi-linear map complex conjugate #\sigma:W\to W# becomes a linear map #\left.W\right|_{\mathbb{R}}\to\left.W\right|_{\mathbb{R}}# with the property that #V# is the eigenspace of #\sigma# at eigenvalue #1# and #\ii V# the eigenspace of #\sigma# at eigenvalue #-1#.

As we will see *later, *it is possible to find a unique matrix in every conjugation class of real #(n\times n)#-matrices, starting with the complex Jordan normal form. However, with the theorem below, it is easy to find out when two real #(n\times n)#-matrices are conjugate, which is the case when they are conjugate as complex matrices.

Complex criterium for real conjugateIf two real #(n\times n)#-matrices are conjugate over #\mathbb{C}#, then they are also conjugate over #\mathbb{R}#.

The matrices #T# and #D# are not real. If we regard #L_A#, #L_D# and #L_T# as linear transformations of #\mathbb{C}^2#, then the eigenvectors (the columns of #T#) each span a complex invariant subspace of #\mathbb{C}^2# of dimension #1#. By taking the real and imaginary part of the first eigenvector #\matrix{5 \\ 2 \complexi-1 \\ }#, we find a basis for the original real vector space #\mathbb{R}^2# relative to which #L_A# has a real matrix. Determine that matrix.

By restricting scalars, we can view the linear map #L_A:\mathbb{C}^2\to \mathbb{C}^2# as a linear map #\left.\left(L_D\right)\right|_{\mathbb{R}}:U\to U#, where #U =\left.\mathbb{C}^2\right|_{\mathbb{R}}#.

As a basis for #V# we choose the real and imaginary part of the eigenvector #\matrix{5 \\ 2 \complexi-1 \\ }# with eigenvalue #1-2 \complexi#:

\[\Re \matrix{5 \\ 2 \complexi-1 \\ } = \matrix{5 \\ -1 \\ } \phantom{xxx}\text{ and }\phantom{xxx} \Im \matrix{5 \\ 2 \complexi-1 \\ } =\matrix{0 \\ 2 \\ }\] The corresponding transition matrix is #S=\matrix{5 & 0 \\ -1 & 2 \\ }#. The matrix of #A# relative to said basis is

\[\begin{array}{rcl} {S}^{-1}\,A\,S &=& {\matrix{{{1}\over{5}} & 0 \\ {{1}\over{10}} & {{1}\over{2}} \\ }}\, \matrix{0 & -5 \\ 1 & 2 \\ }\, \matrix{5 & 0 \\ -1 & 2 \\ }\\

&=& \matrix{1 & -2 \\ 2 & 1 \\ }\end{array}\]

\[\begin{array}{rcl} {\overline{S}}^{-1}\,A\,\overline{S} &=& {\matrix{{{1}\over{5}} & 0 \\ -{{1}\over{10}} & -{{1}\over{2}} \\ }}\, \matrix{0 & -5 \\ 1 & 2 \\ }\, \matrix{5 & 0 \\ -1 & -2 \\ }\\

&=& \matrix{1 & 2 \\ -2 & 1 \\ }\end{array}\] As the notation of the overline above #S# suggests, this basis is the complex conjugate of the columns of #S#.

The matrices #{S}^{-1}\,A\,S# and #{\overline{S}}^{-1}\,A\,\overline{S}# are conjugate by means of #\matrix{0&1\\ 1&0}#:

\[ \matrix{1 & 2 \\ -2 & 1 \\ } =\matrix{0&1\\ 1&0} \matrix{1 & -2 \\ 2 & 1 \\ } \matrix{0&1\\ 1&0} = \matrix{0&1\\ 1&0}^{-1} \matrix{1 & -2 \\ 2 & 1 \\ } \matrix{0&1\\ 1&0}\] The matrices #S^{-1} A S# and #{\overline{S}}^{-1}\,A\,\overline{S}# are the real #(2\times2)#-matrices with respect to the basis #\basis{1,\ii}# that belong to complex multiplication by #1-2 \complexi# and #2 \complexi+1#, respectively.

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