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.