By H. Araki, R. V. Kadison

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SKEW A X E S K x a 2. on x1 x2, a 6, a = XI b + x2 cos y = x,siny FIG. 2. Skew axes in two-dimensional euclidean space. 54 PRODUCT 11. x, 1 I" by = a2 + b2 = xI2 + 2x,x, cos 'p + xi2 c0s7 K = (cosy 1 K. K by on by by do. no a (3) no a priori no priori. do odd on As (3), E-I 5. 55 SKEW AXES by rotational 1 K = -1( 0 1 2 1 0 (37) K f, . by by fn = fo parametric harmonics fo , + nfp , n = on 0, fn ... . by En In fn e = col(... E-, i P = = . ) diag( ... ) 56 11. THE INNER PRODUCT x = = O P (P 0 ) (43) by z, s =x O P e t ~= x (et, it)(P O i )( ) (44) K, by 6.

By by fn = fo parametric harmonics fo , + nfp , n = on 0, fn ... . by En In fn e = col(... E-, i P = = . ) diag( ... ) 56 11. THE INNER PRODUCT x = = O P (P 0 ) (43) by z, s =x O P e t ~= x (et, it)(P O i )( ) (44) K, by 6. ORTHOGONALITY by Definition. Given an inner product relation defined by the metric K, two vectors x and y are orthogonal, or precisely, K-orthogonal, when (x, y) = xtKy = 0 q~ = 0, q~ = Definition. Given an inner product relation defined by the metric K, the set of vectors uiis said to be orthogonal,or better, K-orthogonal, when (uiuj) = u,tKuj = 0 so that each pair of vectors is K-orthogonal.

23 COMMUTATIVITY n x n n x n by A n2 n x n B F n x n cy /3 (cuA + PB) F, F, of A B AB. associative algebra. x(yz) # (xy)~. n x n of 1 Eij A A A = = (aij) r)~ijEij ii on Eij no cu C C ~ , ,=E0~ ~ ij 11. COMM UTATlVlTY F ;i n x n (57) 24 I . VECTORS AND MATRICES AB # BA. 5 1 = 21. 40 53 (52 69) never do n x n commutative subspace n x n n x n on 2 of up no As 12. 25 OF A(B + C) AB AC, (A + B). As (A + B)' = (A + B)(A + B) = A' + BA + AB + B2 (59) no BA AB a 12. DIVISORS O F Z E R O by 1 0 0 0 0 0 (0 O N 1 0 ) = (0 0) by nilpotent.