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LagrangeEuler
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If matrix has integer entries and it is hermitian, are then eigenvalues also integers? Is there some theorem for this, or some counter example?
LagrangeEuler said:If matrix has integer entries and it is hermitian, are then eigenvalues also integers? Is there some theorem for this, or some counter example?
LagrangeEuler said:From ##det(A-\lambda I)=0##. Polynomial with integer coefficients does not need to have integer roots. So I suppose that this is not the case. But here matrices are symmetric so I am not sure. :)
Eigenvalues are a mathematical concept used in linear algebra to describe the behavior of a linear transformation on a vector space. They represent the scalar values that, when multiplied by a given vector, result in a new vector that is parallel to the original vector.
To calculate eigenvalues, you first need to find the characteristic polynomial of the matrix. This is done by taking the determinant of the matrix minus lambda (λ) times the identity matrix. The eigenvalues are then the values of λ that make this polynomial equal to zero.
Eigenvalues are significant because they provide important information about the properties of a matrix. They can be used to determine whether a matrix is invertible, to find eigenvectors, and to solve systems of linear equations. They also have applications in many fields, such as physics, engineering, and computer science.
The eigenvalues of a matrix are determined by its entries. Specifically, the eigenvalues are the roots of the characteristic polynomial, which is composed of the entries of the matrix. Therefore, changing the entries of a matrix can alter its eigenvalues and ultimately its behavior.
Eigenvalues have a wide range of applications in various fields. In physics, they are used to describe the behavior of quantum mechanical systems. In engineering, they are used to analyze structures and systems. In data analysis, they are used in principal component analysis to identify patterns and reduce data dimensionality. In computer graphics, they are used to transform objects in 3D space. These are just a few examples of how eigenvalues are used in real-world applications.