the conjugate eigenvalue or coneigenvalue of the linear transformation. The coneigenvectors and coneigenvalues represent essentially the same information and meaning as the regular eigenvectors and eigenvalues, but arise when an alternative coordinate system is used. The corresponding equation is For example, in coherent electromagnetic scattering theory, the linear transformation represents the action performed by the scattering object, and the eigenvectors represent polarization states of the electromagnetic wave. In optics, the coordinate system is defined from the wave's viewpoint, known as the Forward Scattering Alignment (FSA), and gives rise to a regular eigenvalue equation, whereas in radar, the coordinate system is defined from the radar's viewpoint, known as the Back Scattering Alignment (BSA), and gives rise to a coneigenvalue equation. Generalized eigenvalue problem A generalized eigenvalue problem (second sense) is the problem of finding a (nonzero) vector that obeys where and are matrices. If obeys this equation, with some , then we call the generalized eigenvector of and (in the second sense), and is called the generalized eigenvalue of and (in the second sense) which corresponds to the generalized eigenvector . The possible values of must obey the following equation If linearly independent vectors can be found, such that for every , , then we define the matrices and such that Then the following equality holds And the proof is And since is invertible, we multiply the equation from the right by its inverse, finishing the proof. The set of matrices of the form , where is a complex number, is called a pencil; the term matrix pencil can also refer to the pair of matrices. If is invertible, then the original problem can be written in the form which is a standard eigenvalue problem. However, in most situations it is preferable not to perform the inversion, but rather to solve the generalized eigenvalue problem as stated originally. This is especially important if and are Hermitian matrices, since in this case is not generally Hermitian and important properties of the solution are no longer apparent. If and are both symmetric or Hermitian, and is also a positive-definite matrix, the eigenvalues are real and eigenvectors and with distinct eigenvalues are -orthogonal (). In this case, eigenvectors can be chosen so that the matrix defined above satisfies or , and there exists a basis of generalized eigenvectors (it is not a defective problem). This case is sometimes called a Hermitian definite pencil or