By Alexander Grothendieck
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Extra info for Cohomologie Locale Des Faisceaux Coherents (Sga 2)
By interchanging the diagonal entries like in Step 2 if necessary, we may assume that all the positive entries come ﬁrst. This shows that A is congruent to a diagonal matrix G = D(d1 , . . , d s , -d s+1 , . . , -d r , 0, . , 0), where di > 0. , , 0, . . , 0 , Ë d1 ¯ dr then HGHT has the desired form. To see why s is uniquely determined see [Fink72]. 11 is that it is constructive. 12. Example. To show that the matrix Ê Á1 0 Á A = Á0 0 Á Á0 - 1 Ë 2 ˆ 0 ˜ 1˜ - ˜ 2 ˜ 0 ˜¯ is congruent to a diagonal one with ±1s or 0 on the diagonal.
Proof. 6. Deﬁnition. Let v be a vector. The orthogonal projection of v on X is the orthogonal projection of v on X0. The orthogonal complement of v with respect to X is the orthogonal complement of v with respect to X0. 7, the orthogonal projection of a vector on a plane and its orthogonal complement is well deﬁned. 6 to compute them. A related deﬁnition is Deﬁnition. A vector is said to be parallel to a plane if it lies in the subspace spanned by any basis for the plane. A vector is said to be orthogonal to a plane if it is orthogonal to all vectors in any basis for the plane.
Vk. 18. 6. Proposition. ai = Di . D Proof. See [BoeP94]. Finally, barycentric coordinates are useful in describing linear maps between simplices. 18. Barycentric coordinates and volume ratios. D1 D0 w D2 v0 v1 of another simplex t. Let s = v0v1 · · · vk and t = w0w1 · · · ws. If we express points of s in terms of the (unique) barycentric coordinates with respect to its vertices, then f induces a well-deﬁned map f :sÆ t deﬁned by ˆ k Ê k f Á Â a i v i ˜ = Â a if (v i ). ¯ i =0 Ë i =0 Deﬁnition.