Equivalence (matrix)
The equivalence in the mathematical subfield of linear algebra is an equivalence relation on the class of - matrices .
Two matrices and are by definition equivalent if there is a linear map
- there and there are bases of and of such that
- and
- applies
d. H. Fig. 13 is an illustration of in relation to the bases of and from , and Fig. 6 is an illustration of in relation to the bases of and from .
Equivalent statement
The following statement is equivalent to the statement "the matrices and are equivalent over the body ":
- There is an invertible matrix and an invertible matrix over such that .
Statements about equivalent matrices
- Two regular matrices of the same type are equivalent.
- Two matrices of the same type and rank are equivalent.
Equivalent matrices and similar matrices
A special case of equivalent matrices are the similar matrices .
literature
- Gerd Fischer: Analytical Geometry . 4th edition. Vieweg, 1985, ISBN 3-528-37235-4 . P. 101 and p. 163
See also
Web links
- Eric W. Weisstein : Equivalent Matrix . In: MathWorld (English).