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  "body": "\n## Statement\n\nLet $F$ be a [[field|04C5ZBG1WT]], let $n \\ge 1$ and let $M$ be an $n \\times n$ matrix over $F$.\nThen there are finitely many [[transvection matrices|0DTCBCKEJP]] $L_1, \\dots, L_r$ and $R_1, \\dots, R_s$ and a [[diagonal matrix|044JNC7ZAA]] $D$ over $F$ such that\n$$M = L_1 \\cdots L_r \\, D \\, R_1 \\cdots R_s.$$\n\n## Remarks\n\nMultiplying by a transvection on the left adds a multiple of one row to another row; on the right it adds a multiple of one column to another column.\nThe proof is an induction on $n$.\nIf the last row or the last column of $M$ has a nonzero entry, one row or column addition puts a nonzero entry at position $(n, n)$.\nFurther additions of multiples of row $n$ and column $n$ then clear the rest of the last row and the last column.\nThe upper left block of size $n - 1$ is treated by induction.\nExample: $\\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ becomes $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ after adding column $2$ to column $1$, and then $\\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 0 \\end{pmatrix}$ after subtracting column $1$ from column $2$.\nEach transvection has [[determinant|00KDZRW4C0]] $1$, so $\\det M = \\det D$.\nA refinement of the argument shows that an invertible $M$ is a product of transvections and the matrix $\\operatorname{diag}(1, \\dots, 1, \\det M)$.\nHence the transvections generate the [[special linear group|0V9PMN0RBH]] $\\mathrm{SL}_n(F)$.\n\n## Explanations\n\n### L1 human\n\nUsing only the move \"add a multiple of one row (or column) to another\", every square matrix over a field can be made diagonal.\nNo swaps and no scalings are needed.\nThese moves do not change the determinant, so the diagonal matrix keeps the determinant of the original matrix.\n\n### L2 human\n\nThis is a sharper form of Gaussian elimination: [[elementary matrices|01REXW3WWE]] of the row-addition type already suffice, as long as both row and column operations are allowed.\nThe theorem is the main step in the proof that $\\mathrm{SL}_n(F)$ is generated by transvections.\nIt is also a first step towards the Smith normal form, where the field is replaced by a principal ideal domain and more care is needed.\n",
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