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 . ${\ displaystyle m \ times m}$ ${\ displaystyle S}$ ${\ displaystyle n \ times n}$ ${\ displaystyle T}$ ${\ displaystyle K}$ ${\ displaystyle B = SAT}$