Use the matrices P and D to construct a spectral decomposition of A=PDP 1 1 I √38 19 -5 3-9 - 14 00 6 1 3 -13 -3 P=[4₁ 4₂ 43 ]= 38 0-14 0 05 -9 -3 -5 1 1 3 √√2 √38 √19 A spectral decomposition of A is A = (u₁u+u₂u+uzu, where u₁u₁u₂u, and u-u-. (Simplify your answers.) A = 0 19 D=

Consider the following Gauss-Jordan reduction: Find E₁ = -9 27 1-9 0 1 27-7-3=10001-7-300E₂A-9011 Write A as a product A = E¹E¹E¹E¹ of elementary matrices: 0 11 00-70 E,A >> 010-70 E₂,E,A 0 0 1 0 0 0 1 0 E₂ E₂ E, AO → HEACHEE+BEEHB E3 = 00 01 EEE₂E₁A = | E4= Consider the following Gauss-Jordan reduction Find 9 27 1 0 -7 0 -9 0 27 -7 1 0 A 1 -3 0 - 1 Write A as a product A=E, ¹E, ¹E, ¹E, of elementary matrices 7 00 E₁A 0 0 -7 0 001 100 E₂E₁A 0 0 100 ₂₂A n 00 010-I MESEMBEEHEEEHEEE == 001 EEEEA

Consider the linear transformation T by taking x € R? to T(x) = y in R?, 2.x1 + x2 and y2 = x1 + 2x2. Find an ordered basis so that the corresponding 7.3.12 ND where yı matrix representation of T is diagonal, and find that matrix representation.

2. Let H be the set of 2 × 2 matrices of the form H a b = { (- / + d ) | a,b ≤ c } a where a denotes the complex conjugate of a complex number a. Show that H, together with matrix addition and multiplication, forms a ring. (You may assume that M₂(C) is a ring.)