Inlinear algebra, two rectangularm-by-nmatricesA andB are calledequivalent if

${\displaystyle B=Q^{-1}AP}$

for someinvertiblen-by-n matrixP and some invertiblem-by-m matrixQ. Equivalent matrices represent the samelinear transformationV → W under two different choices of a pair ofbases ofV andW, withP andQ being thechange of basis matrices inV andW respectively.

The notion of equivalence should not be confused with that ofsimilarity, which is only defined for square matrices, and is much more restrictive (similar matrices are certainly equivalent, but equivalent square matrices need not be similar). That notion corresponds to matrices representing the sameendomorphismV → V under two different choices of asingle basis ofV, used both for initial vectors and their images.

Matrix equivalence is anequivalence relation on the space of rectangular matrices.

For two rectangular matrices of the same size, their equivalence can also be characterized by the following conditions

• The matrices can be transformed into one another by a combination ofelementary row and column operations.
• Two matrices are equivalent if and only if they have the samerank.

If matrices are row equivalent then they are also matrix equivalent. However, the converse does not hold; matrices that are matrix equivalent are not necessarily row equivalent. This makes matrix equivalence a generalization of row equivalence.^[1]

Therank property yields an intuitivecanonical form for matrices of the equivalence class of rank${\displaystyle k}$ as

,

where the number of${\displaystyle 1}$s on the diagonal is equal to${\displaystyle k}$. This is a special case of theSmith normal form, which generalizes this concept on vector spaces tofree modules overprincipal ideal domains. Thus:

Theorem: Anymxn matrix of rankk is matrix equivalent to themxn matrix that is all zeroes except that the firstk diagonal entries are ones.^[1]Corollary: Matrix equivalent classes are characterized by rank: two same-sided matrices are matrix equivalent if and only if they have the same rank.^[1]

2x2 matrices only have three possible ranks: zero, one, or two. This means all 2x2 matrices fit into one of three matrix equivalent classes:^[1]

, ,

This means all 2x2 matrices are equivalent to one of these matrices. There is only one zero rank matrix, but the other two classes have infinitely many members; The representative matrices above are the simplest matrix for each class.

Matrix similarity is a special case of matrix equivalence. If two matrices are similar then they are also equivalent. However, the converse is not true.^[2] For example these two matrices are equivalent but not similar:

,

• Row equivalence
• Matrix congruence

• Matrices (mathematics)
• Equivalence (mathematics)

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