Eigenvalues and Eigenvectors - Eigenvalues and Eigenvectors of Matrices - Algebraic and Geometric Multiplicities

Algebraic and Geometric Multiplicities

Given an n×n matrix A and an eigenvalue λ_i of this matrix, there are two numbers measuring, roughly speaking, the number of eigenvectors belonging to λ_i. They are called multiplicities: the algebraic multiplicity of an eigenvalue is defined as the multiplicity of the corresponding root of the characteristic polynomial. The geometric multiplicity of an eigenvalue is defined as the dimension of the associated eigenspace, i.e. number of linearly independent eigenvectors with that eigenvalue. Both algebraic and geometric multiplicity are integers between (including) 1 and n. The algebraic multiplicity n_i and geometric multiplicity m_i may or may not be equal, but we always have m_i ≤ n_i. The simplest case is of course when m_i = n_i = 1. The total number of linearly independent eigenvectors, N_x, is given by summing the geometric multiplicities

Over a complex vector space, the sum of the algebraic multiplicities will equal the dimension of the vector space, but the sum of the geometric multiplicities may be smaller. In this case, it is possible that there may not be sufficient eigenvectors to span the entire space – more formally, there is no basis of eigenvectors (an eigenbasis). A matrix is diagonalizable by a suitable choice of coordinates if and only if there is an eigenbasis; if a matrix is not diagonalizable, it is said to be defective. For defective matrices, the notion of eigenvector can be generalized to generalized eigenvectors, and over an algebraically closed field a basis of generalized eigenvectors always exists, as follows from Jordan form.

The eigenvectors corresponding to different eigenvalues are linearly independent, meaning, in particular, that in an n-dimensional space the linear transformation A cannot have more than n eigenvalues (or eigenspaces). All defective matrices have fewer than n distinct eigenvalues, but not all matrices with fewer than n distinct eigenvalues are defective – for example, the identity matrix is diagonalizable (and indeed diagonal in any basis), but only has the eigenvalue 1.

Given an ordered choice of linearly independent eigenvectors, especially an eigenbasis, they can be indexed by eigenvalues, i.e. using a double index, with x_i,j being the j th eigenvector for the i th eigenvalue. The eigenvectors can also be indexed using the simpler notation of a single index x_k, with k = 1, 2, ..., N_x.