math111_logo 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, cA = c A, AB = A B, (AT) = (A)T, A = BB = A.

As a consequence, we have Av = λvA 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 λv is an eigenvector with eigenvalue λ.

Example Consider the linear transformation T(x1, x2) = (x1 + 5x2, - 2x1 + 3x2): R2R2 given by the matrix

A = [ 1 5 ].
-2 3

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

To understand the transformation T on the real euclidean space R2, we write

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

By comparing the real and the imaginery parts of

T(v) = T(u) + iT(w),
λv = (2 + 3i)(u + iw) = (2u - 3w) + i(3u + 2w),

we find

T(u) = 2u - 3w,
T(w) = 3u + 2w.

In other words, the matrix for T with respect to the basis {u, w} of R2 is

B = [ 2 3 ].
-3 2

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

2 + 3i = √13e = √13cosθ + i13sinθ, 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: R2R2 has a conjugate pair of truly complex eigenvalues λ = μ + and λ = μ - , where ν ≠ 0. Let v = u + iw and v = u - iw be the corresponding eigenvectors. Then {u, v} form a basis of R2, and the matrix of T with respect to the basis is

Λ = [ μ ν ].
- ν μ

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


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