a basis. For example, consider the matrix The roots of the characteristic polynomial are the eigenvalues Solving the linear system gives the eigenvectors and while gives that is, for These vectors form a basis of so we can assemble them as the column vectors of a change-of-basis matrix to get: We may see this equation in terms of transformations: takes the standard basis to the eigenbasis, so we have: so that has the standard basis as its eigenvectors, which is the defining property of Note that there is no preferred order of the eigenvectors in changing the order of the eigenvectors in just changes the order of the eigenvalues in the diagonalized form of Application to matrix functions Diagonalization can be used to efficiently compute the powers of a matrix and the latter is easy to calculate since it only involves the powers of a diagonal matrix. For example, for the matrix with eigenvalues in the example above we compute: This approach can be generalized to matrix exponential and other matrix functions that can be defined as power series. For example, defining we have: This is particularly useful in finding closed form expressions for terms of linear recursive sequences, such as the Fibonacci numbers. Particular application For example, consider the following matrix: Calculating the various powers of reveals a surprising pattern: The above phenomenon can be explained by diagonalizing To accomplish this, we need a basis of consisting of eigenvectors of One such eigenvector basis is given by where ei denotes the standard basis of Rn. The reverse change of basis is given by Straightforward calculations show that Thus, a and b are the eigenvalues corresponding to u and v, respectively. By linearity of matrix multiplication, we have that Switching back to the standard basis, we have The preceding relations, expressed in matrix form, are thereby explaining the above phenomenon. Quantum mechanical application In quantum mechanical and quantum chemical computations matrix diagonalization is one of the most frequently applied numerical processes. The basic reason is that the time-independent Schrödinger equation is an eigenvalue equation, albeit in most of the physical situations on an infinite dimensional space (a Hilbert space). A very common approximation is to truncate Hilbert space to finite dimension, after which the Schrödinger equation can be formulated as an eigenvalue problem of a real symmetric, or complex Hermitian matrix. Formally this approximation is founded on the variational principle,