Matrix Algebra by Abadir K.M., Magnus J.R.

By Abadir K.M., Magnus J.R.

Matrix Algebra is the 1st quantity of the Econometric workouts sequence. It includes routines when it comes to direction fabric in matrix algebra that scholars are anticipated to understand whereas enrolled in an (advanced) undegraduate or a postgraduate path in econometrics or facts. The booklet features a finished number of workouts, all with complete solutions. however the publication is not only a suite of workouts; in reality, it's a textbook, even though person who is geared up in a totally diversified demeanour than the standard textbook. the quantity can be utilized both as a self-contained direction in matrix algebra or as a supplementary textual content.

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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.

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