### Complex Diagonalization

##### 2. Complex eigenvalue and eigenvector for real matrix

The complex conjugation **A** of a matrix **A** is obtained by taking the complex conjugations of the entries of **A**. For example, the complex conjugation of

**A** = [ |
`i` |
-`i` |
1 |
], |

-1 |
1 |
`i` |

0 |
0 |
1 + `i` |

is

**A** = [ |
-`i` |
`i` |
1 |
]. |

-1 |
1 |
-`i` |

0 |
0 |
1 -`i` |

It is easy to see that the conjugation operation has the following properties.

**A** + **B** = **A** + **B**, `c`**A** = `c` **A**, `AB` = **A** **B**, (**A**^{T}) = (**A**)^{T}, **A** = **B** ⇔ **B** = **A**.

As a consequence, we have **Av** = `λ`**v** ⇔ **A** **v** = `λ` **v**, where **v** is obtained by taking complex conjugates of the coordinates of **v**. Combined with the following fact (see the realness property)

**A** is a real matrix ⇔ **A** = **A**,

we have

For a *real* matrix, **v** is an eigenvector with eigenvalue `λ` ⇔ **v** is an eigenvector with eigenvalue `λ`.

Example Consider the linear transformation **T**(`x`_{1}, `x`_{2}) = (`x`_{1} + 5`x`_{2}, - 2`x`_{1} + 3`x`_{2}): **R**^{2} → **R**^{2} given by the matrix

In an earlier example, we found eigenvectors **v** = (1 - 3`i`, 2), **v** = (1 + 3`i`, 2) with eigenvalues `λ` = 2 + 3`i`, `λ` = 2 - 3`i`.

To understand the transformation **T** on the real euclidean space **R**^{2}, we write

**v** = **u** + `i`**w**, for **u** = (1, 2) and **w** = (-3, 0).

By comparing the real and the imaginery parts of

**T**(**v**) = **T**(**u**) + `i`**T**(**w**),

`λ`**v** = (2 + 3`i`)(**u** + `i`**w**) = (2**u** - 3**w**) + `i`(3**u** + 2**w**),

we find

**T**(**u**) = 2**u** - 3**w**,

**T**(**w**) = 3**u** + 2**w**.

In other words, the matrix for **T** with respect to the basis {**u**, **w**} of **R**^{2} is

If we further make use of the polar expression of the eigenvalue:

2 + 3`i` = √13`e`^{iθ} = √13cos`θ` + `i`√13sin`θ`, where `θ` = `tan`^{-1}(2/3) ≈ 33.69°,

we find

**B** = √13[ |
cos(-`θ`) |
-sin(-`θ`) |
]. |

sin(-`θ`) |
cos(-`θ`) |

Thus if we use the coordinate system with respect to the basis {(1, 2), (-3, 0)}, then **T** appears to be a rotation by angle -33.69° followed by multiplying √13.

In general, suppose a *real* linear transformation** T**: **R**^{2} → **R**^{2} has a conjugate pair of *truly complex* eigenvalues `λ` = `μ` + `iν` and `λ` = `μ` - `iν`, where `ν` ≠ 0. Let **v** = **u** + `i`**w** and **v** = **u** - `i`**w** be the corresponding eigenvectors. Then {**u**, **v**} form a basis of **R**^{2}, and the matrix of **T** with respect to the basis is

Geometrically, if we use the polar expression: `λ` = `re`^{iθ} = `r`(cos`θ` + `i`sin`θ`), then **T** is a rotation by angle -`θ` and scaling by `r` with respect to the {**u**, **v**}-coordinates.