In the previous part, we studied the geometric meaning of one conjugate pair of complex eigenvalues and eigenvectors. In this part, we consider several such pairs.

We illustrate the situation by considering a real linear transformation ` T`:

**v**_{1}, **v**_{2} = **u**_{2} + `i w`

with corresponding eigenvalues

`λ`_{1}, `λ`_{2} = `μ`_{2} + `iν`_{2}, `λ`_{2} = `μ`_{2} - `iν`_{2}, `λ`_{3} = `μ`_{3} + `iν`_{3}, `λ`_{3} = `μ`_{3} - `iν`_{3}.

First we prove that the *real* vectors **v**_{1}, **u**_{2}, **w**_{2}, **u**_{3}, **w**_{3} form a basis of **R**^{5}. In fact, for *real* numbers `c`_{1}, `a`_{2}, `b`_{2}, `a`_{3}, `b`_{3}, we have

`c`_{1}**v**_{1} + `a`_{2}**u**_{2} + `b`_{2}**w**_{2} + `a`_{3}**u**_{3} + `b`_{3}**w**_{3} = `c`_{1}**v**_{1} + `c`_{2}**v**_{2} + `c`_{2}**v**_{2} + `c`_{3}**v**_{3} + `c`_{3}**v**_{3},

where `c`_{2} = (`a`_{2} - `ib`_{2})/2, `c`_{3} = (`a`_{3} - `ib`_{3})/2. Then we have

`c`_{1}**v**_{1} + `a`_{2}**u**_{2} + `b`_{2}**w**_{2} + `a`_{3}**u**_{3} + `b`_{3}**w**_{3} = **0**

⇒ `c`_{1}**v**_{1} + `c`_{2}**v**_{2} + `c`_{2}**v**_{2} + `c`_{3}**v**_{3} + `c`_{3}**v**_{3} = **0**

⇒ `c`_{1} = `c`_{2} = `c`_{3} = 0 (**v**_{1}, **v**_{2}, **v**_{2}, **v**_{3}, **v**_{3} are complex linearly independent)

⇒ `c`_{1} = `a`_{2} = `b`_{2} = `a`_{3} = `b`_{3} = 0 (complex number = 0 ⇔ real and imaginary parts = 0)

This means that **v**_{1}, **u**_{2}, **w**_{2}, **u**_{3}, **w**_{3} are linear independent vectors. By this result, they must form a basis of **R**^{5}.

Similar to the computation in the example in the previous part (also see the remark after the example), we have

` T`(

Note that the basis {**v**_{1}, **u**_{2}, **w**_{2}, **u**_{3}, **w**_{3}} gives us a direct sum (see later part of the proof of this result) `span`{**v**_{1}}⊕`span`{**u**_{2}, **w**_{2}}⊕`span`{**u**_{3}, **w**_{3}} = **R**^{5}. This gives the geometric interpretation below.

Let ` V` be a finite dimensional

is a real eigenvector of**v**_{i}, and**T**restricts to a scalar multiplication on**T**`span`{}.**v**_{i}+**u**_{j}`i`and**w**_{j}-**u**_{j}`i`is a pair of complex eigenvectors of**w**_{j}, and**T**restricts to a rotation and scalar multiplication on**T**`span`{,**u**_{j}}.**w**_{j}

The scalar multiplication on `span`{` v_{i}`} is by the real eigenvalue associated to

Finally let us consider the discussion from the matrix viewpoint. Let the 5 by 5 matrix ` A` be the usual matrix of

= [Pv_{1} u_{2} w_{2} u_{3} w_{3}], = [D |
λ_{1} |
0 | 0 | 0 | 0 | ]. |

0 | μ_{2} |
ν_{2} |
0 | 0 | ||

0 | -ν_{2} |
μ_{2} |
0 | 0 | ||

0 | 0 | 0 | μ_{3} |
ν_{3} |
||

0 | 0 | 0 | -ν_{3} |
μ_{3} |

Note that ` D` is a block diagonal matrix, obtained by replacing the conjugate pairs of complex eigenvalues in the usual complex diagonal matrix by special 2 by 2 matrices.

[previous topic] [part 1] [part 2] [part 3] [next topic]