eigenvector

However, this makes sense only if the seeking eigenvectors of and are the same.

The eigenvectors are usually normalized, but they need not be.

Let be a square matrix whose columns are those eigenvectors, in any order.

The advantage of divide-and-conquer comes when eigenvectors are needed as well.

For each of the eigenvalues calculated we are going to have an individual eigenvector.

Note that the definition does not depend on any choice of specific eigenvectors.

It is desired to extend the eigenvectors to a basis for N.

This makes some of the mathematics subtle, since there is a separate eigenvector for every point in space.

Thus, the shear axis is always an eigenvector of S.

The eigenvector of with the lowest eigenvalue is known as the ground state.

We could stop here, but it is a good check to use the eigenvectors to diagonalize A.

Its eigenvectors can be taken as a basis for the vector space the group acts upon.

The amount of influence they have at this steady state is their eigenvector centrality.

If the signs are opposite, the intersection of the eigenvectors is a saddle point.

It then projects the data onto the first k eigenvectors of that matrix.

The eigenvectors are the same too, except expressed in the rescaled basis.

The existence of an eigenvector can be shown in at least two ways:

Pure power methods also can converge slowly, even for the first eigenvector.

Any such row vector is called a left eigenvector of .

To generate the system states where is the matrix of eigenvectors for .

A collection of orthonormal eigenvectors are called the principal directions.

It can be shown that the eigenvectors have the following properties:

The principal axes are the lines spanned by the eigenvectors.

The fiber direction is indicated by the tensor's main eigenvector.

In general, there will be many different eigenvalues for which an eigenvector solution exists.