Let be a real vector space of dimension and . Since is not algebraically closed, might have no eigenvalues, and then it cannot be put in triangular or Jordan form. Nevertheless, every real operator has an invariant subspace of dimension or . The approach is to work over , which is algebraically closed, by extending the scalars of .
7.10.1 The Complexification of a Real Vector Space
Definition 7.10.1 (Complexification). Let be a vector space over . The complexification of is the set , whose elements we write as (), with addition and multiplication by complex scalars We identify with . For , the vectors and are its real and imaginary parts.
The scalar multiplication is what one gets by expanding formally with . Every element of is uniquely with , so two elements are equal if and only if their real parts and their imaginary parts are equal. Note that is closed under addition and real scalars, but not under complex scalars: .
Proposition 7.10.2 ( is a complex vector space). With these operations, is a vector space over . If is a basis of over , then (that is, ) is a basis of over . In particular, .
Proof. Under addition, is an abelian group. The scalar axioms are direct computations; for example, associativity: for and , since . The distributive laws and are checked in the same way.
Spanning: if and with , then .
Independence: if , then comparing real and imaginary parts, and , so all . ∎
Example 7.10.3. The complexification of is : the element of , with , corresponds to the vector , and the operations match.
Definition 7.10.4 (Complex conjugation). The complex conjugate of is .
Conjugation is additive, satisfies , and is conjugate-linear: for . Indeed, for , both and equal . A vector lies in if and only if , and , .
7.10.2 Complexifying Linear Maps
Proposition 7.10.5 (The complexification of a linear map). Let be an -linear map between real vector spaces, and define Then:
- is -linear, and it agrees with on .
- commutes with conjugation: .
- , and for every -linear .
- If and are finite-dimensional with bases and , then , where and are regarded as complex bases of and (Proposition 7.10.2). In particular, for an operator , , a polynomial with real coefficients.
Proof. (a) Additivity is clear. For , which is . And .
(b) .
(c) Clearly , and .
(d) If and , then , so the -th columns of the two matrices agree. The characteristic polynomial of an operator is computed from its matrix. ∎
Note. Part (c) says that , is a functor from real to complex vector spaces, in the language of Chapter 8 (Example 8.2.4).
Now let be an operator on a finite-dimensional real space . Then is an operator on a complex vector space, so it is guaranteed to have an eigenvector if (and it has generalized eigenspaces and a Jordan form). Its eigenvectors tell us about .
Proposition 7.10.6 (Eigenvectors of the complexification). Let , where is a real vector space, and let be an eigenvector of with eigenvalue .
- If , then and lie in , and at least one of them is nonzero; so is an eigenvalue of . Conversely, every eigenvalue of is an eigenvalue of . Thus the real eigenvalues of are exactly the eigenvalues of .
- If with , then is an eigenvector of with eigenvalue ; the vectors are linearly independent; and is a two-dimensional -invariant subspace of , with
Proof. In both cases, and , where . Comparing real and imaginary parts,
(a) If , this says and , and are not both zero because . Conversely, an eigenvector of is an eigenvector of with the same eigenvalue, by Proposition 7.10.5(a).
(b) By Proposition 7.10.5(b), , and . Since , the eigenvectors and are linearly independent over (Theorem 7.4.4). As and , and conversely , , the complex spans of and of coincide and are two-dimensional; so are linearly independent over , and in particular over . The displayed formulas show , and the columns of the matrix are the coordinates of and . ∎
Theorem 7.10.7 (Invariant subspaces of real operators). Every operator on a nonzero finite-dimensional real vector space has an invariant subspace of dimension or .
Proof. Let with finite-dimensional over . Then is a nonzero finite-dimensional complex vector space, and is algebraically closed (the Fundamental Theorem of Algebra, Chapter 4), so has an eigenvector (Theorem 7.5.1). If its eigenvalue is real, Proposition 7.10.6(a) gives an eigenvector of , which spans a one-dimensional invariant subspace. Otherwise, Proposition 7.10.6(b) gives a two-dimensional invariant subspace. ∎
Example 7.10.8 (The rotation, revisited). Let act on . It has no eigenvalues, but its complexification (the same matrix acting on ) has the eigenvector with eigenvalue : indeed . Here , , and . The formulas of Proposition 7.10.6 say and , which is correct, and with matrix in the basis . The conjugate is the eigenvector for .
Note. Since has real coefficients, its non-real roots come in conjugate pairs, matching the pairing of eigenvectors. A real eigenvalue of contributes eigenvectors of itself, while each pair of non-real eigenvalues contributes a two-dimensional invariant subspace on which acts by a matrix , a rotation combined with a scaling. Continuing by induction with the induced operator on (Proposition 7.3.5), as in the proof of Theorem 7.5.2, one can put every real operator in block upper-triangular form with diagonal blocks of size or . Complexification returns in Chapter 9, for operators on real inner product spaces, and in Chapter 16, for real representations.