# Equivalence (matrix)

The equivalence in the mathematical subfield of linear algebra is an equivalence relation on the class of - matrices . ${\ displaystyle m \ times n}$

Two matrices and are by definition equivalent if there is a linear map ${\ displaystyle A}$${\ displaystyle B}$

${\ displaystyle f \ colon \ mathbb {K} ^ {n} \ to \ mathbb {K} ^ {m}}$there and there are bases of and of such that${\ displaystyle B_ {1}, B_ {2}}$${\ displaystyle \ mathbb {K} ^ {n}}$${\ displaystyle C_ {1}, C_ {2}}$${\ displaystyle \ mathbb {K} ^ {m}}$
${\ displaystyle A = _ {B_ {1}} M (f) _ {C_ {1}}}$ and
${\ displaystyle B = _ {B_ {2}} M (f) _ {C_ {2}}}$ 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 . ${\ displaystyle A}$${\ displaystyle f}$${\ displaystyle B_ {1}}$${\ displaystyle \ mathbb {K} ^ {n}}$${\ displaystyle C_ {1}}$${\ displaystyle \ mathbb {K} ^ {m}}$${\ displaystyle B}$${\ displaystyle f}$${\ displaystyle B_ {2}}$${\ displaystyle \ mathbb {K} ^ {n}}$${\ displaystyle C_ {2}}$${\ displaystyle \ mathbb {K} ^ {m}}$

## Equivalent statement

The following statement is equivalent to the statement "the matrices and are equivalent over the body ": ${\ displaystyle m \ times n}$${\ displaystyle A}$${\ displaystyle B}$ ${\ displaystyle K}$

• There is an invertible matrix and an invertible matrix over such that .${\ displaystyle m \ times m}$${\ displaystyle S}$ ${\ displaystyle n \ times n}$${\ displaystyle T}$${\ displaystyle K}$${\ displaystyle B = SAT}$

• 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