Question

Suppose an \(n \times n\) matrix \(A\) with real entries has a complex eigenvalue \(\lambda=\alpha+i \beta(\beta \neq 0)\) with associated eigenvector \(\mathbf{x}=\mathbf{u}+i \mathbf{v},\) where \(\mathbf{u}\) and \(\mathbf{v}\) have real components. Show that \(\mathbf{u}\) and \(\mathbf{v}\) are both nonzero.

Short answer

Question: Prove that the real components of the eigenvector of a complex eigenvalue in a matrix are non-zero. Answer: We showed that assuming either of the real components, \(\mathbf{u}\) or \(\mathbf{v}\), to be zero leads to contradictions, indicating that they must both be non-zero. Therefore, the real components of the eigenvector of a complex eigenvalue in a matrix are non-zero.

Step-by-step solution

Use the eigenvalue-eigenvector equation

We are given that \(\lambda = \alpha + i \beta\) is an eigenvalue with associated eigenvector \(\mathbf{x} = \mathbf{u} + i \mathbf{v}\). Let's substitute these into the eigenvalue-eigenvector equation: $$ A(\mathbf{u} + i \mathbf{v}) = (\alpha + i \beta)(\mathbf{u} + i \mathbf{v}) $$

Rewrite the equation

Now, we will group the real parts and imaginary parts on both sides of the equation: $$ A\mathbf{u} + i(A\mathbf{v}) = \alpha\mathbf{u} - \beta\mathbf{v} + i(\alpha\mathbf{v} + \beta\mathbf{u}) $$

Equate the real and imaginary parts

Since the above equation must hold for all complex numbers, we can equate the real and imaginary parts on both sides: $$ A\mathbf{u} = \alpha\mathbf{u} - \beta\mathbf{v} \quad \text{and} \quad A\mathbf{v} = \alpha\mathbf{v} + \beta\mathbf{u} $$

Reach a contradiction if we assume either \(\mathbf{u}\) or \(\mathbf{v}\) to be zero

Let's assume, towards a contradiction, that either \(\mathbf{u}\) or \(\mathbf{v}\) is zero. Case 1: Assume \(\mathbf{u} = \mathbf{0}\). Then, we have the following from our equations above: $$ A\mathbf{v} = \alpha\mathbf{v} \implies (\alpha I - A) \mathbf{v} = \mathbf{0} $$ This means that \(\alpha\) is a real eigenvalue of A, which contradicts the assumption that \(\alpha + i \beta\) is a complex eigenvalue of A. Therefore, we conclude that \(\mathbf{u}\) cannot be zero. Case 2: Assume \(\mathbf{v} = \mathbf{0}\). Using the equations we derived previously, we get: $$ A\mathbf{u} = \alpha\mathbf{u} \implies (\alpha I - A) \mathbf{u} = \mathbf{0} $$ This also means that \(\alpha\) is a real eigenvalue of A, which again contradicts the assumption that \(\alpha + i \beta\) is a complex eigenvalue. Therefore, we conclude that \(\mathbf{v}\) cannot be zero.

Conclusion

Since we've reached contradictions in both cases, it follows that both \(\mathbf{u}\) and \(\mathbf{v}\) must be non-zero.

Key concepts

Eigenvalue-Eigenvector Equation

In linear algebra, the eigenvalue-eigenvector equation is a fundamental concept. It helps us understand certain properties of matrices and vectors. The equation is given by: \[ A\mathbf{x} = \lambda \mathbf{x} \]Here, \( A \) is a square matrix, \( \mathbf{x} \) is an eigenvector, and \( \lambda \) is the corresponding eigenvalue. This equation states that when the matrix \( A \) acts on the eigenvector \( \mathbf{x} \), the output is simply a scaled version of \( \mathbf{x} \), and the scaling factor is the eigenvalue \( \lambda \).
This relationship is crucial because it allows us to break down complex systems into simpler, more manageable parts. Finding the eigenvalues and eigenvectors lets mathematicians and engineers simplify problems, particularly in systems of differential equations, stability analysis, and vibrations in mechanical systems.
Importantly, when you have complex eigenvalues, the corresponding eigenvectors often have complex entries too, and that brings us to our next topic: complex numbers.

Complex Numbers

Complex numbers are numbers that are constructed from a real part and an imaginary part. They are usually written in the form: \[ a + bi \]where \( a \) denotes the real part and \( bi \) denotes the imaginary part, with \( i \) being the imaginary unit defined as \( i = \sqrt{-1} \). You can think of complex numbers as an extension of the real numbers.
They are incredibly useful in mathematics and engineering because they allow for the solution of equations that have no real solutions, like \( x^2 + 1 = 0 \). Complex numbers also come into play when calculating eigenvalues in linear algebra, especially in systems that involve rotations or oscillations.

• The real part: \( a \)
• The imaginary part: \( bi \)
• Typical representation: \( \, a + bi \, \)

For matrices, if you find a complex eigenvalue like \( \lambda = \alpha + i\beta \), the associated eigenvectors have complex entries, leading us to the next concept.

Real and Imaginary Parts

When dealing with complex numbers especially in eigenvectors and eigenvalues, it is crucial to break them into real and imaginary parts. Let's say we have an eigenvalue \( \lambda = \alpha + i\beta \) and a corresponding eigenvector \( \mathbf{x} = \mathbf{u} + i\mathbf{v} \). Here, \( \alpha \) and \( \beta \) are the real and imaginary parts of the eigenvalue, while \( \mathbf{u} \) and \( \mathbf{v} \) represent the real and imaginary parts of the eigenvector.
This breakdown is essential because by separating these components, we can handle complex arithmetic much more easily. For example, when multiplying two complex numbers, you'll distribute them just like binomials but with factoring in \( i^2 = -1 \).
In the step-by-step solution, we managed to show that neither the real part \( \mathbf{u} \) nor the imaginary part \( \mathbf{v} \) of the eigenvector can be zero, which ensures that the properties of the system being analyzed remain complex in nature.

Matrices with Complex Entries

Matrices with complex entries arise naturally when dealing with systems that exhibit complex behavior, particularly those involving complex eigenvalues and eigenvectors. A matrix with complex entries can be denoted as:\[A = \begin{bmatrix} a_{11}+ic_{11} & a_{12}+ic_{12} & \cdots & a_{1n}+ic_{1n} \ a_{21}+ic_{21} & a_{22}+ic_{22} & \cdots & a_{2n}+ic_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{n1}+ic_{n1} & a_{n2}+ic_{n2} & \cdots & a_{nn}+ic_{nn} \\end{bmatrix}\]Each entry can have a real part and an imaginary part, which is crucial in solving various types of linear algebraic problems.
These matrices are especially significant in fields such as quantum mechanics, control theory, and vibration analysis, where complex values represent different states or behaviors. When handling such matrices, it is crucial to understand the role of complex eigenvalues, as they can indicate oscillations or rotational behaviors in systems.
Keep in mind, however, that while working with them, separate computations into real and imaginary components, making analysis more straightforward and computationally manageable.