An eigenvalue is a characteristic value that represents the scale factor of a linear transformation. It is a special value that does not change direction when a transformation is applied to a vector.
An eigenvector is a vector that does not change direction when a linear transformation is applied to it. It is associated with a specific eigenvalue and represents the direction of the transformation.
Finding eigenvalues is important because it allows us to understand the behavior of linear transformations and systems of differential equations. It also helps in solving many problems in physics, engineering, and other fields.
The process for finding eigenvalues involves solving a characteristic equation, which is obtained by setting the determinant of a matrix to zero. The resulting solutions are the eigenvalues of the matrix.
No, not all matrices can be diagonalized. Only square matrices with linearly independent eigenvectors can be diagonalized. Matrices with repeated eigenvalues or complex eigenvalues cannot be diagonalized.