Matrix similarity

The similarity in the mathematical subfield of linear algebra is an equivalence relation on the class of square matrices. Similar matrices describe the same linear map ( endomorphism ) when using different bases.

Two matrices, and on the same body are similar matrices, if there is an invertible matrix over, so that

Or equivalent

Applies.

Homothecy

A mapping that assigns a matrix A a similar one matrix B is called similarity mapping or similarity transformation. We then have (see above).

Diagonalizability, Trigonalisierbarkeit

If a matrix A is similar to a diagonal matrix, then we say that it is diagonalizable. A matrix is trigonalisierbar if it is similar to an upper triangular matrix.

Properties of similar matrices

Similar matrices have the same eigenvalues ​​of (but not necessarily the same eigenvectors ). It follows that it

  • The same rank,
  • The same determinant,
  • The same track,
  • The same characteristic polynomial,
  • The same minimal polynomial and
  • The same Jordan normal form

Have.

Two matrices are then exactly similar if their characteristic matrices are equivalent (so-called Frobenius lemma ).

Generalization

The similarity of matrices is an equivalence relation. It is a special case of the general as defined in the equivalence class of matrices.

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