The eigenvalue quotient of an operator is the quotient of this type, here Lambda is an unknown scholar, phi is unknown vector and "a" is an operator under consideration. To solve the eigenvalue equation means to find all possible values of Lambda for which it can hold and for each such Lambda, to find the set of vectors which satisfy the quotient with this Lambda. All Lambdas satisfying the quotient are called the eigenvalues of the operator, while the vectors corresponding to those Lambdas are called the eigenvectors of the operator. Let's look at some properties of eigenvalues and eigenvectors. First, if you have some eigenvector phi corresponding to some eigenvalue Lambda 1, then for any complex number Alpha, Alpha phi will also be the eigenvector corresponding to this Lambda 1. To eliminate this ambiguity, we can choose to normalize the vectors which are the solutions of this equation. Now suppose that for some eigenvalue, let's called Lambda 2, they are big linearly independent eigenvectors, then it is quite obvious that any vector constructed as a linear combination of these phi eigenvectors is also the solution of the eigenvalue equation is Lambda 2. This means that each eigenvalue defines the subspace of the initial space, and each vector of this subspace is an eigenvector corresponding to this value. Eigenvalue which has more than one eigenvector is called degenerate, the degree of degeneracy of an eigenvalue defines the dimensional analogy of the Eigen subspace. The eigenvectors have the geometrical meaning of some special directions and touched in some sense by the operator. If the operator, for example, rotates the three-dimensional Euclidean space around some axis, then this axis itself would not be touch by such rotation, thus it is going to be an eigenvector of such operator with eigenvalue 1. Now, if this rotation is by angle phi, then the whole plane orthogonal to this excess preserves the direction of its vectors, hey're just multiplied by minus 1 by this rotation. This plane is two-dimensional, which means that eigenvalue minus 1 is two degenerate in this case, and this plane orthogonal to the axis of rotation is two-dimensional eigenspace corresponding to this value. It is quite obvious that two different eigenvalues cannot have mutual eigenvectors, and dimensional space and operator can have at most only n linearly independent eigenvectors. Each such vector corresponds to some eigenvalue, and some of them can correspond to the same degenerate eigenvalue. This whole means that the eigenvalue quotient of an operator in an n-dimensional space can have at most n solutions, n-eigenvalues. The eigenvalue equation is a powerful instrument for operator analysis in general, but from now on, we are going to concentrate on a very special case. We will consider only infinite dimensional spaces, which we deal with bandwidth consider cubits and we are going to examine only Hermitian operators. Now, for Hermitian operators, the solutions of this equation have additional properties. First, it is easy to show that eigenvalues of Hermitian operator are always real, their complex or imaginary part is 0 always. Second, the eigenvectors of different eigenvalues are mutually orthogonal, this is very important property, it means that for a Hermitian operator in infinite dimensional space, we can construct an orthonormal basis of its eigenvectors. Even if some eigenvalues of the operator are degenerate, we still can do it because in the Eigen subspace of a degenerate eigenvalue, we can choose any Eigenvectors we want, so we can choose orthonormal set of vectors there. Each Hermitian operator defines an orthonormal basis, and we remember that each orthonormal basis in the state space defines a set of possible values of the system measurement, so now we can give at last the strict mathematical definition of an observable of a quantum system with infinite dimensional state space and here it is, an observable is a Hermitian operator acting in the states space of a system. The eigenvectors of this operator are the measurement outcomes. They are the states to which the system can collapse after the measurement and when it does, the classical outcome that we obtain as a result of the measurement is the eigenvalue, which corresponds to that eigenvector. This is all good, but wait, we know that some eigenvalues can be degenerate, so if the system collapses to some vector from the eigen subspace of degenerate eigenvalue, we obtain only that eigenvalue and we can tell which vector represents the system now, so we must admit that in this case, the system collapses not to a single vector, but to the whole eigen subspace, it is very true. It is much better if you have an observable with all eigenvalues being non-degenerate, but if you don't have one, we can use a set of several commuting observables. It can be proved and this is a fundamental theorem in quantum mechanics that if two observables commute, one can construct an orthonormal basis of their common eigenvectors, so if you managed to implement a set of commuting observables, such as for each their common eigenvector corresponds a non degenerate eigenvalue of one of them, we will be able to obtain the most possible information about the state of a quantum system using this set of observables. Such sets are called the CSCO, the complete set of commuting observables. However, we are not going to discuss this as you hope further here since most of the observables we are going to use have all non-degenerate Eigenvalues, thus representing the complete set of commuting observables alone. I believe that it is time to observe some examples.