# Question involving eigenvectors

• dingo_d
In summary, the eigenvectors can be real or complex, depending on the context, and the normalization is not trivial.
dingo_d

## Homework Statement

So I have to find the eigenvalues and eigenvectors of

$$A=\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$$

which is not that special and hard. I solve the characteristic equation and find the eigenvalues: $$\lambda_{1,2}=\pm 1$$. So finding the eigenvectors is relatively simple, I just plug that back into characteristic eq and I get:

$$v_1=t\left(\begin{array}{c}1\\1\end{array}\right)$$

and

$$v_2=t\left(\begin{array}{c}1\\-1\end{array}\right)$$.

So that's pretty simple? I can even normalize it and show that they are orthogonal. But!

There was one thing that kinda bugged me, and got my attention. Back on linear algebra class we said that t is some real parameter and we lived our lives happily ever after.

But my professor, now on QM asked this: how do we know that t is real? What if it's complex?

If it is complex then the normalization isn't that trivial.

So my question is: why is it real? Are we free to impose that on the parameter? Or is there some deeper math behind it all?

dingo_d said:

## Homework Statement

So I have to find the eigenvalues and eigenvectors of

$$A=\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$$

which is not that special and hard. I solve the characteristic equation and find the eigenvalues: $$\lambda_{1,2}=\pm 1$$. So finding the eigenvectors is relatively simple, I just plug that back into characteristic eq and I get:

$$v_1=t\left(\begin{array}{c}1\\1\end{array}\right)$$

and

$$v_2=t\left(\begin{array}{c}1\\-1\end{array}\right)$$.

So that's pretty simple? I can even normalize it and show that they are orthogonal. But!

There was one thing that kinda bugged me, and got my attention. Back on linear algebra class we said that t is some real parameter and we lived our lives happily ever after.

But my professor, now on QM asked this: how do we know that t is real? What if it's complex?

If it is complex then the normalization isn't that trivial.

So my question is: why is it real? Are we free to impose that on the parameter? Or is there some deeper math behind it all?
In general, t or your eigenvectors needn't be real. For example, if we let t=i then your eigenvectors are v1 = [i, i]T and v2 = [i, -i]T. Substituting the eigenvectors into the eigenvalue problem yields

$$\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}\begin{bmatrix}i \\ i\end{bmatrix} = \begin{bmatrix}i \\ i\end{bmatrix}$$

and

$$\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}\begin{bmatrix}i \\ -i\end{bmatrix} = -\begin{bmatrix}i \\ -i\end{bmatrix}$$.

Therefore, it is possible in general to have complex eigenvectors. However, if your matrix is a symmetric (as it is here) nxn square matrix, then there exists n mutually orthogonal, real eigenvectors; but that doesn't mean that there aren't any complex ones!

In general, the context dictates whether the eigenvectors should be real, rather than the matrix itself.

And in the context of quantum mechanics? Usually the eigenvectors represent states in which observable has a definite value (the eigenvalue).

What would imaginary eigenvector represent?

dingo_d said:
And in the context of quantum mechanics? Usually the eigenvectors represent states in which observable has a definite value (the eigenvalue).

What would imaginary eigenvector represent?
In quantum mechanics, the eigenvectors (or wave functions) belong to a complex Hilbert Space. In other words, quantum mechanical wave functions are, in general, complex.

Hootenanny said:
In quantum mechanics, the eigenvectors (or wave functions) belong to a complex Hilbert Space. In other words, quantum mechanical wave functions are, in general, complex.

Oh I see! Cool, thanks on the clearing that out ^^

## 1. What are eigenvectors and why are they important in science?

Eigenvectors are special vectors that do not change direction when multiplied by a particular matrix. They are important in science because they provide a way to understand the transformation of a system and its properties.

## 2. How do you find eigenvectors?

Eigenvectors can be found by solving the characteristic equation of a matrix or by using specialized algorithms such as the power method or the QR algorithm.

## 3. What is the significance of eigenvalues in relation to eigenvectors?

Eigenvalues are the corresponding scalars that represent the magnitude of an eigenvector. They provide important information about the behavior of a system and its properties.

## 4. Can eigenvectors be used in real-world applications?

Yes, eigenvectors have a wide range of applications in fields such as physics, engineering, and computer science. They are used in image and signal processing, data compression, and machine learning, among others.

## 5. Are there any limitations to using eigenvectors?

While eigenvectors are a powerful tool for understanding and analyzing systems, they have limitations in certain situations. For example, they may not exist for every matrix, and they may not provide a complete understanding of the system's behavior.

• Introductory Physics Homework Help
Replies
8
Views
817
• Differential Equations
Replies
2
Views
815
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Linear and Abstract Algebra
Replies
5
Views
319
Replies
6
Views
1K
• Introductory Physics Homework Help
Replies
8
Views
1K
• Introductory Physics Homework Help
Replies
13
Views
1K
Replies
13
Views
2K
• Introductory Physics Homework Help
Replies
4
Views
383
• Linear and Abstract Algebra
Replies
14
Views
1K