Generalized Gaussian elimination Order of elimination arbitrary. Set . Select an arbitrary as the first pivot element. We can eliminate from all but the equation. The multipliers are . i1-st The reduced system is now . Select another pivot and repeat the elimination. If , then the remaining equations are degenerate and we halt. Theorem 3.2: Let have rank Then we can find a sequence of distinct row A r. and column indices (i1,j1), (i2,j2), (ir,jr) such that corresponding pivot …, elements in are nonzero and . Define permutation matrices A(1), A(2), A(r) …, (whose columns are unit vectors) and , where {ik} and {jk} are permutations of Then {1,2,…,n}.
Generalized Gaussian elimination Order of elimination arbitrary. Set . Select an arbitrary as the first pivot element. We can eliminate from all but the i1-st equation. The multipliers are . The reduced system is now . Select another pivot and repeat the elimination. If , then the remaining equations are degenerate and we halt. Theorem 3.2: Let A have rank r. Then we can find a sequence of distinct row and column indices (i1,j1), (i2,j2), …, (ir,jr) such that corresponding pivot elements in A(1), A(2), …, A(r) are nonzero and . Define permutation matrices (whose columns are unit vectors) and , where {ik} and {jk} are permutations of {1,2,…,n}. Then
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