Determinantal Variety
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High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! Given m and n and r min(m, n), the determinantal variety Y r is the set of all m × n matrices (over a field k) with rank r. This is naturally an algebraic variety as the condition that a matrix have rank r is given by the vanishing of all of its (r + 1) × (r + 1) minors. Considering the generic m × n matrix whose entries are algebraically independent variables x i,j, these minors are polynomials of degree r + 1. The ideal of k[x i,j] generated by these polynomials is a determinantal ideal. Since the…mehr

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High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! Given m and n and r min(m, n), the determinantal variety Y r is the set of all m × n matrices (over a field k) with rank r. This is naturally an algebraic variety as the condition that a matrix have rank r is given by the vanishing of all of its (r + 1) × (r + 1) minors. Considering the generic m × n matrix whose entries are algebraically independent variables x i,j, these minors are polynomials of degree r + 1. The ideal of k[x i,j] generated by these polynomials is a determinantal ideal. Since the equations defining minors are homogeneous, one can consider Y r either as an affine variety in mn-dimensional affine space, or as a projective variety in (mn 1)-dimensional projective space.