Examples of "eigenbasis"
Dividing these by their respective lengths yields an orthonormal eigenbasis:
By the Karhunen–Loève theorem, one can express the centered process in the eigenbasis,
where the "c" are the coordinates with respect to the given eigenbasis. Furthermore,
The quantum state at each instant can be expressed as a linear combination of the complete eigenbasis of formula_4:
An important example of this is Fourier analysis, which diagonalizes the heat equation using the eigenbasis of sinusoidal waves.
With these definitions it is easy to describe the process of collapse. For any observable, the wave function is initially some linear combination of the eigenbasis formula_9 of that observable. When an external agency (an observer, experimenter) measures the observable associated with the eigenbasis formula_10, the wave function "collapses" from the full formula_11 to just "one" of the basis eigenstates, formula_12, that is:
In its simplest form it is expressed as a unitary transformation relating the flavor and mass eigenbasis, and can be written
To orthogonally diagonalize "A", one must first find its eigenvalues, and then find an orthonormal eigenbasis. Calculation reveals that the eigenvalues of "A" are
The proof is easy: consider an eigenbasis for . The basis in can be indexed by sequences , indeed, consider the symmetrizations of
where formula_14 is the formula_15th eigenpair after orthonormalization and formula_16 is the formula_15th coordinate of "x" in the eigenbasis. It is then easy to verify that the bounds are attained at the corresponding eigenvectors formula_18.
Consider a system prepared in state formula_15. Since the eigenstates of the observable formula_11 form a complete basis called eigenbasis, the state vector formula_15 can be written in terms of the eigenstates as
Consider a system prepared in state formula_22. Since the eigenstates of the observable formula_11 form a complete basis called eigenbasis, the state vector formula_15 can be written in terms of the eigenstates as
In particular, "A" is "orthogonally diagonalizable", since one may take a basis of each eigenspace and apply the Gram-Schmidt process separately within the eigenspace to obtain an orthonormal eigenbasis.
For the second part, suppose that the eigenvalues of "A" are λ, ..., λ (possibly repeated according to their algebraic multiplicities) and the corresponding orthonormal eigenbasis is u...,u. Then
Now suppose Alice is an observer for system , and Bob is an observer for system . If in the entangled state given above Alice makes a measurement in the formula_19 eigenbasis of , there are two possible outcomes, occurring with equal probability:
then it is called a ";" this corresponds to there being a common eigenbasis (a basis of simultaneous eigenvectors) for all the represented elements of the algebra, i.e., to their being simultaneously diagonalizable matrices (see diagonalizable matrix).
The difference between the two expressions ["I"("ρ") − "J"("ρ")] defines the basis-dependent quantum discord, which is asymmetrical in the sense that formula_5 can differ from formula_6. The notation "J" represents the part of the correlations that can be attributed to classical correlations and varies in dependence on the chosen eigenbasis; therefore, in order for the quantum discord to reflect the purely nonclassical correlations independently of basis, it is necessary that "J" first be maximized over the set of all possible projective measurements onto the eigenbasis:
Any subspace spanned by eigenvectors of "T" is an invariant subspace of "T", and the restriction of "T" to such a subspace is diagonalizable. Moreover, if the entire vector space "V" can be spanned by the eigenvectors of "T", or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of "T" is the entire vector space "V", then a basis of "V" called an eigenbasis can be formed from linearly independent eigenvectors of "T". When "T" admits an eigenbasis, "T" is diagonalizable.
Suppose a state formula_26 is a state in the simultaneous eigenbasis of formula_27 and formula_28 (i.e., a state with a single, definite value of formula_27 and a single, definite value of formula_28). Then using the commutation relations, one can prove that formula_31 and formula_32 are "also" in the simultaneous eigenbasis, with the same value of formula_27, but where formula_34 is increased or decreased by formula_24, respectively. (It is also possible that one or both of these result vectors is the zero vector.) (For a proof, see ladder operator#angular momentum.)
Such a transformation is called a diagonalizable matrix since in the eigenbasis, the transformation is represented by a diagonal matrix. Because operations like matrix multiplication, matrix inversion, and determinant calculation are simple on diagonal matrices, computations involving matrices are much simpler if we can bring the matrix to a diagonal form. Not all matrices are diagonalizable (even over an algebraically closed field).