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mathboy
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I know that if T has eigenvalue k, then T* has eigenvalue k bar. But if T has eigenvector x, does T* also have eigenvector x? If so, how do you prove it? I don't see that in my textbook.
morphism said:What about the converse? Namely, if T and T* share their eigenvectors, is T necessarily normal?
Eigenvalues and eigenvectors are concepts in linear algebra that are used to analyze the properties of a linear transformation or a matrix. Eigenvalues represent the scaling factor of the eigenvectors when the transformation is applied, and they can reveal important information about the transformation, such as its stability or invertibility.
Proving the existence of eigenvalues and eigenvectors is important because it allows us to understand the behavior of a linear transformation or a matrix. It can help us determine the stability of a system, find the optimal solution to a problem, or analyze the properties of a matrix. Additionally, eigenvalues and eigenvectors are used in various fields of science, including physics, engineering, and computer science.
To prove eigenvalues and eigenvectors for T and T*, you can use various methods such as the characteristic polynomial method, the diagonalization method, or the power method. These methods involve finding the characteristic polynomial of the matrix, solving for its roots, and then finding the corresponding eigenvectors.
Eigenvalues and eigenvectors have many applications in various fields. In physics, they are used to represent the energy levels of quantum systems. In engineering, they are used to analyze the stability of structures. In computer science, they are used in data compression and image processing. They are also used in statistics, finance, and many other areas of science and technology.
Yes, there are some limitations to the methods used for proving eigenvalues and eigenvectors. For example, the characteristic polynomial method may not always work for matrices with complex eigenvalues. Additionally, the power method may not converge or may converge to a wrong eigenvector if the matrix is not diagonalizable. It is important to choose the appropriate method depending on the properties of the matrix and the desired results.