Similarity (matrix)
In the mathematical subfield of linear algebra , similarity is an equivalence relation on the class of square matrices . Similar matrices describe the same linear self-mapping ( endomorphism ) when using different bases.
definition
Two- dimensional square matrices over the body are called similar to each other if there is a regular matrix such that
or equivalent
applies. The image
is called similarity mapping or similarity transformation . If a matrix is similar to a diagonal matrix , it is said to be diagonalizable ; if it is similar to an upper triangular matrix , it is called trigonalizable .
example
The two real matrices
- and
are similar to each other because with the regular matrix
applies
- .
The matrix is not unique, because each multiples with meets these identity.
properties
Parameters
Two matrices that are similar to one another have the same characteristic polynomial , because it holds with the commutativity of the identity matrix , the determinant product theorem and the determinant of the inverse
Therefore have matrices similar to one another
- the same eigenvalues (but not necessarily the same eigenvectors ),
- the same determinant and
- the same trace .
They also have matrices that are similar to one another
- the same rank ,
- the same minimal polynomial and
- the same Jordanian normal form .
characterization
Two complex matrices are similar to each other if and only if they have the same Jordanian normal form (except for the order of the Jordan blocks).
In general, according to the lemma of Frobenius two matrices and if and similar to each other if they have the same Frobenius normal form possess. This is the case if and only if their characteristic matrices and have the same Smith normal form .
Equivalence classes
The similarity of matrices is an equivalence relation , i.e. reflexive , symmetric and transitive . One writes
- ,
when and are similar to each other and recorded to a matrix corresponding equivalence class by
- .
For example, there is the equivalence class of up to a multiple of the identity matrix -like arrays of exactly one element , because regular for all matrices .
The similarity of matrices is a special case of the more generally defined equivalence on the class of matrices.
Calculation of the transformation matrix
method
Given two matrices that are similar to one another , a matrix with which applies can be determined as follows. First, the two matrices and are converted into the same Frobenius normal form (or, if possible, the same Jordan normal form) . Are the two similarity transformations used for this
- and
with regular matrices , it follows by equating
- .
The transformation matrix we are looking for is therefore
- .
example
Let the two matrices and be given as in the example above. The characteristic polynomials of the two matrices result in
and
- .
The two characteristic polynomials therefore agree, with the eigenvalues being and . Because the characteristic polynomial completely breaks down into real linear factors, the same Jordan normal form can be set up for both matrices, which in this case is the diagonal form
Has. The transformation matrices into the Jordan normal form have the form and , with eigenvectors for the eigenvalue and eigenvectors for the eigenvalue . For there are two eigenvectors by solving and as
- and .
Correspondingly, for two eigenvectors, by solving and as
- and .
The two transformation matrices into the Jordan normal form are accordingly
- and ,
and the similarity transformation matrix we are looking for is thus
- .
See also
literature
- Gerd Fischer : Linear Algebra. 18th edition. Springer Spectrum, 2014, ISBN 978-3-8348-0996-4 .