Complex Eigenvalues

Consider the linear homogeneous system

displaymath221

The Characteristic polynomial is

displaymath223

In this section, we consider the case when the above quadratic equation has complex roots (that is if tex2html_wrap_inline225 ). The roots (eigenvalues) are

displaymath227

where

displaymath229

In this case, the difficulty lies with the definition of

displaymath231

In order to get around this difficulty we use Euler's formula

displaymath233

Therefore, we have

displaymath235

In this case, the eigenvector associated to tex2html_wrap_inline237 will have complex components.

Example. Find the eigenvalues and eigenvectors of the matrix

displaymath239

Answer. The characteristic polynomial is

displaymath241

Its roots are

displaymath243

Set tex2html_wrap_inline245 . The associated eigenvector V is given by the equation tex2html_wrap_inline249 . Set

displaymath251

The equation tex2html_wrap_inline249 translates into

displaymath255

Since tex2html_wrap_inline257 , then the two equations are the same (which should have been expected, do you see why?). Hence we have tex2html_wrap_inline259 which implies that an eigenvector is

displaymath261

We leave it to the reader to show that for the eigenvalue tex2html_wrap_inline263 , the eigenvector is

displaymath265

Let us go back to the system

displaymath221

with complex eigenvalues tex2html_wrap_inline269 . Note that if V, where

displaymath273

is an eigenvector associated to tex2html_wrap_inline275 , then the vector

displaymath277

(where tex2html_wrap_inline279 is the conjugate of v) is an eigenvector associated to tex2html_wrap_inline283 . On the other hand, we have seen that

displaymath285

are solutions. Note that these solutions are complex functions. In order to find real solutions, we used the above remarks. Set

displaymath287

then we have

displaymath289

which gives

displaymath291

Similarly we have

displaymath293

Putting everything together we get

displaymath295

Clearly this implies tex2html_wrap_inline297 where

displaymath299

It is easy to see that we have

displaymath301

Since the sum and difference of solutions lead to another solution, then both tex2html_wrap_inline303 and tex2html_wrap_inline305 are solutions of the system. These are real solutions. It is very easy to check in fact that they are linearly independent. Let us summarize the above technique.

Summary (of the complex case). Consider the system

displaymath221

tex2html_wrap_inline309
Write down the characteristic polynomial

displaymath223

and find its roots

displaymath313

we are assuming that tex2html_wrap_inline225 . Note that at this step, you need to know tex2html_wrap_inline317 and tex2html_wrap_inline319 . The common mistake is to forget to divide by 2.

tex2html_wrap_inline309
Find an eigenvector V associated to the eigenvalue tex2html_wrap_inline275 . Write down the eigenvector as

displaymath327

tex2html_wrap_inline309
Two linearly independent solutions are given by the formulas

displaymath299

tex2html_wrap_inline309
The general solution is

displaymath335

where tex2html_wrap_inline337 and tex2html_wrap_inline339 are arbitrary numbers. Note that in this case, we have

displaymath341

Example. Consider the harmonic oscillator

displaymath343

Find the general solution using the system technique.

Answer. First we rewrite the second order equation into the system

displaymath345

The matrix coefficient of this system is

displaymath239

We have already found the eigenvalues and eigenvectors of this matrix. Indeed the eigenvalues are

displaymath349

Hence we have

displaymath351

The eigenvector associated to tex2html_wrap_inline237 is

displaymath355

Next we write down the two linearly independent solutions

displaymath357

and

displaymath359

The general solution of the equivalent system is

displaymath335

or

displaymath423

Below we draw some solutions. Notice how the solutions spiral and dye at the origin (see the discussion below)

Since we are looking for the general solution of the differential equation, we only consider the first component. Therefore we have

displaymath363

You may want to check that the second component is just the derivative of y.
Below we draw some solutions for the differential equation

Qualitative Analysis of Systems with Complex Eigenvalues.

Recall that in this case, the general solution is given by

displaymath341

The behavior of the solutions in the phase plane depends on the real part tex2html_wrap_inline317 . Indeed, we have three cases:

tex2html_wrap_inline309
the case: tex2html_wrap_inline14 . The solutions tend to the origin (when tex2html_wrap_inline437 ) while spiraling. In this case, the equilibrium point is called a spiral sink.

tex2html_wrap_inline309
The case: tex2html_wrap_inline441 The solutions explode or get away from the origin (when tex2html_wrap_inline437 ) while spiraling. In this case, the equilibrium point is called a spiral source.

tex2html_wrap_inline309
The case: tex2html_wrap_inline16 The solutions are periodic. This means that the trajectories are closed curves or cycles. In this case, the equilibrium point is called a center.

If you would like more practice, click on Example.

[Differential Equations] [First Order D.E.]
[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

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Author: Mohamed Amine Khamsi

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