An eigenvector is a vector that does not change direction when multiplied by a matrix. In other words, it is a special vector that remains in the same direction after a transformation. In the context of A_n, an eigenvector is a vector that, when multiplied by the matrix A_n, yields a scalar multiple of itself. This scalar multiple is known as the eigenvalue.
To find the eigenvector and eigenvalue of A_n, you can follow these steps:
Yes, A_n can have multiple eigenvectors for a given eigenvalue. In fact, there can be an infinite number of eigenvectors for a single eigenvalue. This is because any scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue.
To show that a vector v is an eigenvector of A_n, you can multiply v by A_n and see if the result is a scalar multiple of v. In other words, if Av = λv, where λ is a scalar, then v is an eigenvector of A_n with eigenvalue λ.
Yes, A_n can have complex eigenvalues and eigenvectors. This is because the characteristic polynomial of A_n can have complex roots, which will result in complex eigenvalues. The corresponding eigenvectors will also be complex. However, A_n can also have real eigenvalues and eigenvectors, depending on the matrix's properties.