By R.A. Kalnin
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B) If AB and BA are both defined, that is, if n = p and m = q, they may not be of the same order. For example, if A has order 2 × 3 and B has order 3 × 2 then AB has 24 2 Matrices order 2 × 2 and BA has order 3 × 3. If AB and BA are both defined and are of the same order, then both must be square. Even in that case AB = BA is general. For example, the matrices 0 1 0 1 A= and B = 1 0 −1 0 do not commute. The matrix A swaps the rows of B when it premultiplies it, while swapping the columns of B when it postmultiplies it.
B) Any vector x ⊥ b must satisfy x, b = 0, that is, x1 = 0. Hence, x=λ 0 1 (λ ∈ R) . (c) If x ⊥ y, then x, y = 0 and hence, x+y 2 = x + y, x + y = x, x + 2 x, y + y, y = x, x + y, y = x 2 + y 2. 13 (Orthonormal vectors) Two orthogonal vectors x and y that are normalized such that x = y = 1 are said to be orthonormal. (a) Show that the unit vectors ei are orthonormal. (b) If x := m i=1 ci ei , determine the values of the ci . (c) Discuss the geometric meaning of this result. Solution (a) This follows from the fact that ei , ei = 1 and ei , ej = 0 (i = j).
Solution (a) This is essentially a generalization of the fact that any normalized real 2 × 1 vector x has a representation x = (cos θ, sin θ) . Let a c A := b . d The equations A A = AA = I yield a2 + b2 = 1, a2 + c2 = 1, b2 + d2 = 1, ab + cd = 0, ac + bd = 0, c2 + d2 = 1, and implying a2 = d 2 , b2 = c2 , a2 + b2 = 1, ab + cd = 0. This gives a = cos θ, b = − sin θ, c = ± sin θ, d = ± cos θ. The matrix A1 rotates any vector x := (x, y) by an angle θ in the positive (counterclockwise) direction.