What do the eigenvalues represent?
The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the eigensystem of that transformation. The set of all eigenvectors of T corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace or characteristic space of T.
Eigenspace. If is an square matrix and is an eigenvalue of , then the union of the zero vector and the set of all eigenvectors corresponding to eigenvalues is a subspace of known as the eigenspace of . SEE ALSO: Eigen Decomposition, Eigenvalue, Eigenvector.
- Diagonalizable matrix. Diagonalization is the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map. A square matrix that is not diagonalizable is called defective.
- Definition: the algebraic multiplicity of an eigenvalue e is the power to which (λ – e) divides the characteristic polynomial. Definition: the geometric multiplicity of an eigenvalue is the number of linearly independent eigenvectors associated with it. That is, it is the dimension of the nullspace of A – eI.
- Characteristic polynomial. In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix as coefficients.
Matrix diagonalization is the process of taking a square matrix and converting it into a special type of matrix--a so-called diagonal matrix--that shares the same fundamental properties of the underlying matrix. Similarly, the eigenvectors make up the new set of axes corresponding to the diagonal matrix.
- The diagonalization theorem states that an matrix is diagonalizable if and only if has linearly independent eigenvectors, i.e., if the matrix rank of the matrix formed by the eigenvectors is . All normal matrices are diagonalizable, but not all diagonalizable matrices are normal.
- Linear Combination of Eigenvectors is Not an Eigenvector Suppose that and are two distinct eigenvalues of a square matrix and let and be eigenvectors corresponding to and , respectively. Express as a Linear Combination Determine whether the following set of vectors is linearly independent or linearly dependent.
- Eigenspace. If is an square matrix and is an eigenvalue of , then the union of the zero vector and the set of all eigenvectors corresponding to eigenvalues is a subspace of known as the eigenspace of . SEE ALSO: Eigen Decomposition, Eigenvalue, Eigenvector. CITE THIS AS: Weisstein, Eric W. "
In essence, an eigenvector v of a linear transformation T is a non-zero vector that, when T is applied to it, does not change direction. Applying T to the eigenvector only scales the eigenvector by the scalar value λ, called an eigenvalue. This condition can be written as the equation.
- In graph theory, betweenness centrality is a measure of centrality in a graph based on shortest paths. The betweenness centrality for each vertex is the number of these shortest paths that pass through the vertex.
- Definition of betweenness. : the quality or state of being between two others in an ordered mathematical set.
- Betweenness of Points By definition, a point B is between two other points A and C if all three points are collinear and AB + BC = AC. Although this definition is unambiguous and easy to state, it is not always easy to work with in proofs, because we may not always know what the distances AB, BC, and AC are.
Updated: 17th October 2019