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  "body": "\n## Definition\n\nLet $F$ be a field and let $A$ and $B$ be $m \\times n$ matrices over $F$.\nThe matrix $B$ is *equivalent* to $A$ when there are an [[invertible|0EB9N0S33C]] $n \\times n$ matrix $P$ and an invertible $m \\times m$ matrix $Q$ over $F$ with\n$$B = Q^{-1} A P.$$\n\n## Remarks\n\nTwo matrices are equivalent if and only if they represent the same linear map $F^n \\to F^m$ in two pairs of bases.\nTwo $m \\times n$ matrices over a field are equivalent if and only if they have the same [[rank|0ZC9YTXZRY]].\nA matrix of rank $r$ is equivalent to the block matrix $\\begin{pmatrix} I_r \u0026 0 \\\\ 0 \u0026 0 \\end{pmatrix}$.\n[[Similarity|0E89HQATM8]] is a finer relation: similar matrices are equivalent, but equivalent square matrices need not be similar.\n\n## Explanations\n\n### L1 human\n\nMatrix equivalence allows independent changes of basis on the input side and on the output side.\nWith so much freedom, only the rank survives.\nEvery matrix of rank $r$ becomes the identity of size $r$, padded with zeros.\n",
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