eigendecomposition

It is also possible to find the solution by finding the eigendecomposition of a larger system.

Comment: This is not a special case of the eigendecomposition (see above).

Computing the eigendecomposition is an expensive operation, especially as increases.

Thus the matrix required for the eigendecomposition of is .

This is equivalent to the eigendecomposition of :

This reduces the necessary eigendecomposition to just .

The eigendecomposition allows for much easier computation of power series of matrices.

If is the eigendecomposition of , then .

Eigendecomposition of a symmetric matrix (decomposition according to the spectral theorem)

Now consider an eigendecomposition of , with unit-norm eigenvectors and corresponding eigenvalues :

This decomposition is called the eigendecomposition of , and it is preserved under change of coordinates.

Equivalently, the same computation is performed by diagonalization of A through use of its eigendecomposition:

Fault detection of NCS based on eigendecomposition, adaptive evaluation and adaptive threshold.

Principal component scores are computed through an eigendecomposition of a sample's covariance matrix and rotates the data to preserve procrustes distances.

The advantage of this is that the eigenvalues of A can be controlled by setting K appropriately through eigendecomposition of .

Comment: The eigendecomposition is useful for understanding the solution of a system of linear ordinary differential equations or linear difference equations.

Giving us the solutions of the eigenvalues for the matrix as or , and the resulting diagonal matrix from the eigendecomposition of is thus .

Another approach uses the Nyström method to approximate the eigendecomposition of the Gram matrix K, using only a random sample of the training set.

Of course, this creates linearly independent vectors, so there is no covariance on which to perform eigendecomposition explicitly as we would in linear PCA.

When eigendecomposition is used on a matrix of measured, real data, the inverse may be less valid when all eigenvalues are used unmodified in the form above.

To compute S(ρ) it is convenient (see logarithm of a matrix) to compute the Eigendecomposition of .

The discrete heat kernel requires eigendecomposition of a matrix of size , where is the number of vertices in the mesh representation of the manifold.

The eigendecomposition of a symmetric positive semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue.

Comment: the Jordan normal form generalizes the eigendecomposition to cases where there are repeated eigenvalues and cannot be diagonalized, the Jordan-Chevalley decomposition does this without choosing a basis.

The spectral theorem also provides a canonical decomposition, called the spectral decomposition, eigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts.