# A History of Geometrical Methods by Julian Lowell Coolidge

By Julian Lowell Coolidge

Full, authoritative background of the suggestions for facing geometric equations covers improvement of projective geometry from old to trendy instances, explaining the unique works, commenting at the correctness and directness of proofs, and displaying the relationships among arithmetic and different highbrow advancements. 1940 edition.

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OΓ e v e r Therefore y 9 which is not majorized by p. This proves (2). (3) can be proved similarly. Lemma 2. | { Mp,p 6 Pn,f(p) < k},{Up,p 6 Pn,((p) < k) are bases of Vn,k Proof. , f(p) > k. Let / E Vn}k. ,Xk) (7) Therefore any / £ VJjjj. can be written as a linear combination of Λίp's for which p 6 Λi,^(p) < k. ,xk)=*0. , £(p), (note ί(p) < k) we obtain ap = 0. 1 Majorization ordering 45 in Vn /ς. This shows that { Mp,p G Pn, ί(p) < k} is a basis of V^ j.. To show that {Up,p ζ. Pn,t{p) < k} is a basis it suffices to observe q

For a positive definite A let A? Γf where Γ*is orthogonal and D is diagonal in A=ΓDΓf. Now the conditional distribution of A2W\A2 given W— /is the same as the conditional distribution of W\ given W = A. This follows from the above mentioned fact that the conditional distribution does not depend on Σ. Therefore (12) SUpiAWό I W= 1} = ^ ( W Ί ) I W= A}. Letting A = W\ + W2 we obtain from (11) and (12) (13) ε{yP(wi) I Wi + w2) = cpyv{wx + w2). 4 Coefficients oflίq Lemma 4. 57 Let W\ have the density (10).

Since (5) corresponds to the uniquely defined Gram-Schmidt orthonormalization T, H are unique. Now W = UU9 = ΊT9 is distributed according to ^(Ikyk). 3. To show (i) and (ii) we first note that for any orthogonal Γ, UΓ has the same distribution as U. Furthermore UΓ = T(HΓ). Therefore HΓ is the resulting orthogonal matrix obtained by performing Gram-Schmidt orthonormalization to the rows of UΓ and T is common to U and UΓ. This implies that given T the conditional distributions of H and HΓ are the same.