Lecture 30 - Complexification

We've been overlooking the problem of having $V$ be a real vector space. We want to know what properties can carry over from the complex to the real vector spaces. For example, waht can we say about operators on a real vector space? What about characteristic and minimal polynomials?

The idea here is to have $V$ be in a subspace of some higher, complex vector space $V_{C}$ . This concept is called complexification. Also given an operator $T \in L (V)$ , we can define an associated map $T_{C} \in L (V_{C})$ to agree with $T$ at least on $V$ .

complexification of $V$ , $V_{C}$

Suppose $V$ is a real vector space.

The complexification of $V$ , denoted $V_{C}$ , equals $V \times V$ . An element of $V_{C}$ is an ordered pair $(u, v)$ where $u, v \in V$ , but we'll write this as $u + i v$ .
Addition on $V_{C}$ is defined by:
$(u_{1} + i v_{1}) + (u_{2} + i v_{2}) = (u_{1} + u_{2}) + i (v_{1} + v_{2})$
for $u_{1}, v_{1}, u_{2}, v_{2} \in V$ .
Complex scalar multiplication on $V_{C}$ is defined by:
$(a + b i) (u + i v) = (a u - b v) + i (a v + b u)$
for $a, b \in R$ and $u, v \in V$ .

You can see that $V = {(v, 0) : v \in V} \subseteq V \times V = V_{C}$ .

complexification of $T$ , $T_{C}$

Suppose $V$ is a real vector space and $T \in L (V)$ . The complexification of $T$ , denoted $T_{C}$ , is the operator $T_{C} \in L (V_{C})$ defined by:
$T_{C} (u + i v) = T u + i T v$
for $u, v \in V$ .

Minimal and Characteristic Polynomials of $T_{C}$

Notice that:

Minimal polynomial of

T_{C}

equals minimal polynomial of

T

Suppose $V$ is a real vector space and $T \in L (V)$ . Then the minimal polynomial of $T_{C}$ equals the minimal polynomial of $T$ .

comes from:

$(T_{C})^{n} (u + i v) = T^{n} u + i T^{n} v$

So then extending this to polynomials:

p (T_{C}) = (p (T))_{C}

for all $p \in P (R)$ . Let $μ_{T}$ be the minimal polynomial of $T$ , so then:

μ_{T} (T_{C}) = (μ_{T} (T))_{C}

If $q$ is a monic polynomial that has $q (T_{C}) = \vec{0}$ then is it possible for $\deg μ_{T} > \deg q$ ? It's not possible because we can write:

q = r + i s

where $r, s \in P (R)$ . And because $\deg r = \deg q$ since $q$ is monic (and thus the largest degree coefficient is real and goes to $r$ ) then since $q (T_{C}) = \vec{0}$ then $r (T_{C}) = (r (T))_{C} = \vec{0}$ so then $r (T) = \vec{0}$ so then $\deg r \geq \deg μ_{T}$ by the definition of $μ_{T}$ being the minimal polynomal. So then $\deg q \geq μ_{T}$ .

As a result:

μ_{T} = μ_{T_{C}}

So then $μ_{T_{C}}$ is always going to have real coefficients, even though $V_{C}$ is a complex vector space!

Eigenvalues of $T_{C}$

Some cool facts:

Real eigenvalues of

T_{C}

Suppose $V$ is a real vector space, $T \in L (V)$ and $λ \in R$ . Then $λ$ is an eigenvalue of $T_{C}$ iff $λ$ is an eigenvalue of $T$ .

Nonreal eigenvalues of

T_{C}

come in pairs

Suppose $V$ is a real vector space, $T \in L (V)$ and $λ \in C$ . Then $λ$ is an eigenvalue of $T_{C}$ iff $\overset{―}{λ}$ is an eigenvalue of $T_{C}$ .

Multiplicity of

λ

equals the multiplicity of

\overset{―}{λ}

Suppose $V$ is a real vector space, $T \in L (V)$ and $λ \in C$ is an eigenvalue of $T_{C}$ . Then the multiplicity of $λ$ as an eigenvalue of $T_{C}$ equals the multiplicity of $\overset{―}{λ}$ as an eigenvalue of $T_{C}$ .

Minimal and Characteristic Polynomials of TC

Eigenvalues of TC

Minimal and Characteristic Polynomials of $T_{C}$

Eigenvalues of $T_{C}$