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ELLIPSES - ECONOMETRIC THEORY Exercise 2 Matrix...

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ECONOMETRIC THEORY: Exercise 2 Matrix Multiplication: Orthonormal Matrices and Rotations. 1. Find following matrix product: S 1 0 0 S 2 ‚ • 1 ρ ρ 1 ‚ • S 1 0 0 S 2 . 2. Confirm that, if θ = 45 , then cos θ - sin θ sin θ cos θ = " 1 2 - 1 2 1 2 1 2 # . 3. Use the definitions of a sine and a cosine together with Pythagorus theorem to prove that sin 2 + cos 2 = 1. 4. Confirm that cos θ sin θ - sin θ cos θ ‚ • cos θ - sin θ sin θ cos θ = 1 0 0 1 and that cos θ - sin θ sin θ cos θ ‚ • cos θ sin θ - sin θ cos θ = 1 0 0 1 . 5. Calculate the product " 1 2 1 2 - 1 2 1 2 # 1 - ρ 0 0 1 + ρ " 1
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