Given a nilpotent square matrix A of order n over an algebraically closed field F, the following algorithm produces an invertible matrix C and a Weyr matrix W such that W = C ^ {-1 } AC.
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All the book has presented so far are matrices as systems of equations, square matrices as products of elementary matrices, a simple version of the invertible matrix theorem, the definition of determinants, and Cramer's rule.
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In mathematics, the "'general linear group "'of degree " n " is the set of group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible.
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The group is so named because the columns of an invertible matrix are linearly independent, hence the vectors / points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.
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By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix " L " and an upper triangular matrix " U " if and only if all its leading principal minors are non-zero.
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Under a change of basis, the column " x " is multiplied on the left by an invertible matrix " S ", and the symmetric square matrix " A " is transformed into another symmetric square matrix " B " of the same size according to the formula
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By left-multiplication with an appropriate invertible matrix " L ", it can be achieved that row " t " of the matrix product is the sum of ? times the original row " t " and ? times the original row " k ", that row " k " of the product is another linear combination of those original rows, and that all other rows are unchanged.
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