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  "body": "\n## Definition\n\nLet $R$ be a ring and let $n \\ge 0$ be an integer.\nA *square matrix* of order $n$ over $R$ is an $n \\times n$ [[matrix|0VD9ZSWWJG]] over $R$.\nIts *main diagonal* is the list of the entries $a_{11}, a_{22}, \\dots, a_{nn}$.\nThe set of square matrices of order $n$ over $R$ is written $M_n(R)$.\n\n## Remarks\n\nThe product of two $n \\times n$ matrices is again an $n \\times n$ matrix, so $M_n(R)$ is a [[ring|0FHVPRAE60]].\nThe [[trace|0XC1RDBY7A]], the [[determinant|00KDZRW4C0]] and the [[eigenvalues|0BQAA2BK3Q]] are defined for square matrices.\nA square matrix represents a linear map from a space to itself.\n\n## Explanations\n\n### L1 human\n\nA square matrix has as many rows as columns, like a $3 \\times 3$ table.\nOnly square matrices can be multiplied by themselves, so only they have powers like $A^2$.\nMany important notions, such as the determinant, apply only to square matrices.\n",
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