Question

Ist es möglich, für eine reelle \(2 \times 2\) Matrix einen reellen und einen komplexen Eigenwert zu erhalten?

Short answer

Nein, es ist nicht möglich, für eine reelle \(2 \times 2\) Matrix einen reellen und einen komplexen Eigenwert zu erhalten. Die Eigenwerte sind entweder beide reell oder konjugierte komplexe Paare.

Step-by-step solution

Verständnis der Eigenschaften der Eigenwerte reeller Matrizen

Zuerst sollten wir in Betracht ziehen, dass die Eigenwerte einer Matrix die Lösungen des Eigenwertproblems sind, das durch die Gleichung \(Ax = \lambda x\) definiert ist, wobei A die Matrix ist, \(\lambda\) der Eigenwert und x der Eigenvektor ist. Für reelle \(2 \times 2\) Matrizen können die Eigenwerte entweder reell oder komplex sein, aber sie kommen immer in konjugierten Paaren vor, wenn sie komplex sind.

Bewertung der Fragenaufgabe

Gegeben die oben genannten Eigenschaften von Eigenwerten, würde die Frage darauf hinauslaufen, ob es möglich ist, einen reellen und einen komplexen Eigenwert für die gleiche \(2 \times 2\) Matrix zu haben. Da jedoch reelle Eigenwerte einzeln und komplexe Eigenwerte paarweise vorkommen, ist es nicht möglich, für eine reelle \(2 \times 2\) Matrix einen reellen und einen komplexen Eigenwert zu erhalten. Die Eigenwerte sind entweder beide reell oder konjugierte komplexe Paare.

Key concepts

$2 \times 2$ Matrizen

In mathematics, a matrix is a rectangular array of numbers arranged in rows and columns. A \(2 \times 2\) matrix is one with two rows and two columns. These matrices form the basis of many problems in linear algebra. They can represent simple linear transformations in a 2-dimensional space, like rotations or scaling. The structure of a \(2 \times 2\) matrix looks like this:

• \[A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\]

This simple form makes them a critical topic when learning about eigenvalues. They are small enough to solve by hand but complex enough to demonstrate important mathematical principles. Linear equations and transformations involving these matrices are solvable using methodical approaches, like solving for eigenvectors and eigenvalues.

komplexe Zahlen

Complex numbers are numbers that have a real part and an imaginary part. They take the form \(a + bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the imaginary unit with the property that \(i^2 = -1\). These numbers expand the set of real numbers and are essential in many areas of advanced mathematics. When dealing with eigenvalues of matrices,

• complex numbers help in describing rotations and oscillations.
• For real matrices, complex eigenvalues occur in conjugate pairs; if \(\lambda = a + bi\) is an eigenvalue, so is \(\overline{\lambda} = a - bi\).

This property is crucial for matrices used in physics and engineering, allowing for complex transformations in systems like quantum mechanics.

Eigenwertproblem

The eigenvalue problem is central to linear algebra and consists of finding the eigenvalues \(\lambda\) and eigenvectors \(x\) for a matrix \(A\). The equation \(Ax = \lambda x\) expresses this relationship.

• The main idea is to find values \(\lambda\) such that the matrix \(A - \lambda I\) has no inverse, meaning the determinant \(\text{det}(A - \lambda I) = 0\).
• This results in a characteristic polynomial that can be solved to give the eigenvalues.

In the context of \(2 \times 2\) matrices, the characteristic polynomial is quadratic, yielding two solutions. These solutions are the eigenvalues, which can be real or complex. This process is fundamental in understanding how linear transformations scale different vector spaces.

konjugierte Paare

Conjugate pairs are a fundamental concept in the context of complex numbers. When matrices have complex eigenvalues, these values occur in conjugate pairs, especially for real matrices. If a \(2 \times 2\) real matrix has eigenvalues, they can be either two real numbers or a pair of complex conjugates.

• Complex conjugates are pairs of complex numbers \(a + bi\) and \(a - bi\).
• Their imaginary parts are equal in magnitude but opposite in sign, effectively balancing each other.

This symmetry is important in ensuring that the solutions to the eigenvalue problem for real matrices remain consistent. Hence, it is impossible for a real \(2 \times 2\) matrix to have one real and one complex eigenvalue, as complex eigenvalues must exist in conjugate pairs.