Linear mapping/Diagonalizable/Eigentheory/Section

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The restriction of a linear mapping to an eigenspace is the homothety with the corresponding eigenvalue, so this is a quite simple linear mapping. If there are many eigenvalues with high-dimensional eigenspaces, then usually the linear mapping is simple in some sense. An extreme case are the so-called diagonalizable mappings.

For a diagonal matrix

the characteristic polynomial is just

If the number occurs -times as a diagonal entry, then also the linear factor occurs with exponent inside the factorization of the characteristic polynomial. This is also true when we just have an upper triangular matrix. But in the case of a diagonal matrix, we can also read of immediately the eigenspaces, see example. The eigenspace for consists of all linear combinations of the standard vectors , for which equals . In particular, the dimension of the eigenspace equals the number how often occurs as a diagonal element. Thus, for a diagonal matrix, the algebraic and the geometric multiplicities coincide.

Let denote a field, let denote a vector space, and let

denote a linear mapping. Then is called diagonalizable, if has a basis consisting of eigenvectors

for .

Let denote a field, and let denote a finite-dimensional vector space. Let

denote a

linear mapping. Then the following statements are equivalent.
  1. is diagonalizable.
  2. There exists a basis of such that the describing matrix is a diagonal matrix.
  3. For every describing matrix with respect to a basis , there exists an invertible matrix such that

    is a diagonal matrix.

The equivalence between (1) and (2) follows from the definition, from example, and the correspondence between linear mappings and matrices. The equivalence between (2) and (3) follows from fact.

Let denote a field, and let denote a finite-dimensional vector space. Let

denote a linear mapping. Suppose that there exists different eigenvalues. Then is diagonalizable.

Because of fact, there exist linearly independent eigenvectors. These form, due to fact, a basis.

We continue with example. There exists the two eigenvectors and for the different eigenvalues and , so that the mapping is diagonalizable, due to fact. With respect to the basis , consisting of these eigenvectors, the linear mapping is described by the diagonal matrix

The transformation matrix, from the basis to the standard basis , consisting of and , is simply

The inverse matrix is

Because of fact, we have the relation