An introduction to algebra and geometry via matrix groups by Boij M., Laksov D.

By Boij M., Laksov D.

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Sets with a distance function enjoying the properties of the proposition appear everywhere in mathematics. It is therefore advantageous to axiomatize their properties. 10. Let X be a set. A metric on X is a function d: X × X → R such that, for any triple x, y, z of points of X, we have (i) d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y, (ii) d(x, y) = d(y, x), (iii) d(x, z) ≤ d(x, y) + d(y, z). 11. For every subset Y of a metric space (X, dX ), we have a distance function dY on Y defined by dY (x, y) = dX (x, y), for all x and y in Y .

M and all other ei , ej = 0. With respect to this basis, we have that m x, y = (ai bn+1−i − an+1−i bi ). i=1 It follows from the proposition that all non-degenerate alternating bilinear forms on a vector space are equivalent. 8. 6 is called a symplectic basis. A linear map α : V → V such that α(x), α(y) = x, y , for all pairs x, y of V , is called symplectic. The set of all symplectic linear maps is denoted by Sp(V, , ). 1 we see that Sp(V, , ) is a subgroup of Gl(V ), We call the group Sp(V, , ) the symplectic group, of , .

Consequently it suffices to prove assertion (i) for diagonal matrices. 12 (iv) that a1 ··· log exp exp a1 ··· 0 .. . . . 0 ··· an = log .. 0 .. log exp a1 ··· 0 .. ··· exp an = .. 0 .. 0 .. ··· log exp an a1 ··· = 0 .. . . . 0 ··· an Hence we have proved the first assertion. To prove assertion (ii) we use that exp log A and A are continuous functions from U to Mn (C). Reasoning as in the proof of assertion (i) we see that it suffices to prove assertion (ii) for diagonal matrices. The verification of the assertion for diagonal matrices is similar to the one we used in the proof for diagonal matrices in assertion (i).

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