How is an eigenvector related to a particular matrix?

How is an eigenvector related to a particular matrix?
When you multiply one of a matrix's eigenvectors by the matrix, you get a scalar multiple of that vector back. There are finitely many such vectors and they are specific to a particular matrix. (The scalar multiples are called the eigenvalues.)
An eigenvector, for a linear transformation is a nonzero vector such that for some scalar called an eigenvalue.Originally Posted by BlueLantern
So if is an eigenvalue and is an associated eigenvector then . This shows that has a nonzero kernel and therefore is not invertible. From that one concludes that relative to any basis the matrix for must have zero determinant. The converse is also true so that the eigenvalues of the linear transfromation are the solutions of the equation . This equation is called the characteristic equation for and the polynomial in , , is called the characteristic polynomial.
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