By Abadir K.M., Magnus J.R.
Matrix Algebra is the 1st quantity of the Econometric routines sequence. It includes workouts when it comes to path 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 includes a finished choice of routines, all with complete solutions. however the ebook is not only a set of routines; in reality, it's a textbook, even though one who is geared up in a totally varied 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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For each (n − 1) × (n − 1) permutation matrix there are precisely n ways to form an n × n permutation matrix. Hence, there exist n! permutation matrices of order n. (d) Each row pi. of the permutation matrix P contains one 1 and (n − 1) zeros. Hence, pi. pi. = 1. Another row, say pj . , also contains only one 1, but in a different place. Hence, pi. pj . = 0 (i = j). Thus P is orthogonal. 30 (Normal matrix) A real square matrix A is normal if A A = AA . (a) Show that every symmetric matrix is normal.
20 (Inner product in Cm ) The inner product between two complex vec∗ tors u and v is the complex number u, v := m i=1 ui vi and the norm is the real number 1/2 m u := u, u 1/2 = ( i=1 ui u∗i ) . Show that for any vectors u, v, w in Cm and any scalar λ in C: (a) u, v = v, u ∗ ; (b) u, v + w = u, v + u, w ; (c) λu, v = λ u, v ; (d) u, u ≥ 0, with u, u = 0 ⇐⇒ u = 0. Solution (a) u, v = (b) We have ∗ i u i vi = ∗ ∗ i (vi ui ) = v, u ∗ . ui (vi + wi )∗ = u, v + w = i i ui vi∗ + = i ui (vi∗ + wi∗ ) ui wi∗ = u, v + u, w .
B) x = x, x 1/2 ≥ 0, with x = 0 ⇐⇒ x, x = 0 ⇐⇒ x = 0. 9) gives x, y x+y 2 2 ≤ x 2 y 2 . Hence, = x + y, x + y = x, x + 2 x, y + y, y ≤ x 2 +2 x · y + y 2 = ( x + y )2 . Taking the square root of both sides yields the triangle inequality. Equality occurs if and only if x, y = x · y , that is, if and only if x and y are collinear (Cauchy-Schwarz). The geometric interpretation of the inequality is that in any triangle, the sum of the lengths of two sides must exceed the length of the third side. 1.