Why do antisymmetric eigenvalues have to be purely imaginary?

Thread starter SeReNiTy
Start date May 16, 2006
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Eigenvalues

In summary, Antisymmetric eigenvalues are a specific type of eigenvalues in a matrix that have properties such as always coming in pairs, having a sum of zero, and being used in various areas of mathematics and physics. They can be calculated using methods such as the characteristic polynomial method and have applications in real-world problems such as stability analysis and machine learning.

May 16, 2006

SeReNiTy

Why do they have to purely imaginary?

I got a proof that looks like Ax=ax
where a = eigenvalue

therefore Ax.x = ax.x = a|x|^2

Ax.x = x.(A^t)x
where A^t = transpose = -A
x.(-A)x = -b|x|^2

therefore a=-b, where b = conjugate of a

Now is this as far as i need to go?

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May 16, 2006

matt grime

Science Advisor

Homework Helper

Write it as a=x+iy, b=x-iy, now you're saying what...

FAQ: Why do antisymmetric eigenvalues have to be purely imaginary?

What is the definition of antisymmetric eigenvalues?

Antisymmetric eigenvalues refer to a specific type of eigenvalues in a matrix, where the elements of the matrix are equal to the negative of their transposes. This means that the matrix is symmetric with respect to the main diagonal and has all its main diagonal elements equal to zero.

What are the properties of antisymmetric eigenvalues?

The properties of antisymmetric eigenvalues include:

They always come in pairs, with opposite signs.
The sum of the eigenvalues is always equal to zero.
The product of the eigenvalues is always equal to the determinant of the matrix.
The eigenvalues are always imaginary or zero.
The eigenvectors corresponding to the eigenvalues are orthogonal to each other.

How are antisymmetric eigenvalues used in mathematics and physics?

Antisymmetric eigenvalues are used in various areas of mathematics and physics, including:

Quantum mechanics, where they are used to represent observable quantities such as energy and angular momentum.
Differential equations, where they are used to solve systems of equations with periodic solutions.
Graph theory, where they are used to analyze the structure of networks.
Control theory, where they are used in the design of control systems.

How can one calculate the antisymmetric eigenvalues of a matrix?

The antisymmetric eigenvalues of a matrix can be calculated using various methods, such as:

The characteristic polynomial method, where the eigenvalues are the roots of the characteristic polynomial of the matrix.
The diagonalization method, where the eigenvalues are the diagonal elements of the diagonalized matrix.
The trace method, where the eigenvalues are equal to the trace of the matrix.
The power method, where the eigenvalues can be estimated by repeatedly multiplying the matrix by a vector and normalizing the result.

What are the applications of antisymmetric eigenvalues in real-world problems?

Antisymmetric eigenvalues have applications in various real-world problems, such as:

Stability analysis of physical systems, where the eigenvalues are used to determine the stability of the system.
Signal processing, where the eigenvalues are used to analyze signals and extract information.
Image processing, where the eigenvalues are used for image compression and feature extraction.
Machine learning, where the eigenvalues are used to reduce the dimensionality of data and improve the efficiency of learning algorithms.