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Leo321
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We have vectors x,y of size n and a matrix A of size nxn.
Is it true that the matrix xyTA has at most one non zero eigenvalue? Why is it so?
Is it true that the matrix xyTA has at most one non zero eigenvalue? Why is it so?
khemist said:xy(transpose) will yield a scalar correct?
Eigenvalues and eigenvectors are mathematical concepts used in linear algebra. Eigenvalues are scalar values that represent how a linear transformation changes a vector. Eigenvectors are the corresponding vectors that, when multiplied by the transformation, result in a scalar multiple of themselves.
Eigenvalues and eigenvectors are used in various scientific fields, including physics, engineering, and computer science. They are useful in solving systems of linear equations, analyzing data, and understanding the behavior of complex systems.
The process of calculating eigenvalues and eigenvectors involves finding the eigenvectors of a matrix by solving a characteristic equation. This equation is derived from the matrix and involves finding the values that make the determinant of the matrix equal to zero. These values are the eigenvalues, and the corresponding eigenvectors can be found using the eigenvalue-eigenvector equation.
Eigenvalues and eigenvectors play a crucial role in understanding linear transformations and their effects on vectors. They also have practical applications in solving systems of linear equations and diagonalizing matrices, making them essential tools in linear algebra.
Yes, eigenvalues and eigenvectors can have complex values. In fact, in many cases, complex eigenvalues and eigenvectors provide more useful information about the behavior of a system than real ones. In quantum mechanics, for example, complex eigenvalues and eigenvectors are used to describe the wave functions of particles.