HW4_602_2011

HW4_602_2011 - Math 602 Homework 4 1 Consider the quadratic form on R3 x2 − 2x2 x2 4x1 x2 8x1 x3 4x2 x3 = F(x 1 2 3 Find the following a an

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Unformatted text preview: Math 602 Homework 4 1. Consider the quadratic form on R3 x2 − 2x2 + x2 + 4x1 x2 + 8x1 x3 + 4x2 x3 = F (x) 1 2 3 Find the following: a) an orthonormal basis of unit vectors {e1 , e2 , e3 }, and a set {λ1 , λ2 , λ3 } so that 3 3 2 λi yi yi ei ⇒ F (x) = x= 1 1 3 b) Find an orthogonal matrix U such that x = U y ⇒ F (x) = 1 3 c) Find the max and min of F (x) subject to i=1 2 λi yi . x2 = 1. i 2 −1 −1 2 Verify that the matrix R you found is indeed positive semidefinite and that R 2 = A. 2. Calculate the square root R of the matrix A = 3. Let a, k be constants, with a > 0. Calculate ∞ 2 +kx e−ax dx −∞ by completing the square and using the fact that ∞ −y 2 −∞ e dy = √ π. 4. Consider the integral ∞ ∞ J≡ e− ··· −∞ x,Ax + k,x dx1 · · · dxn −∞ where x, Ax is a(n arbitrary) positive definite quadratic form, and k is an (arbitrary) constant vector. (i) With the help of coordinates which diagonalize the quadratic form, show that the integral J can be brought to the form ∞ ∞ ··· J= −∞ e− y,Λy + B k,y dy1 · · · dyn −∞ where Λ is a diagonal matrix (which one?), and B is a suitable matrix (which one?). (Hint: unitary transformations preserve lengths, angles, hence volumes and the element of volume: if x = U y with U unitary then dx1 · · · dxn = dy1 · · · yn .) 1 (ii) By shifting the coordinates y = (y1 , · · · , yn ) by a vectorial amount b, y =w+b bring the exponential under the integral into a form with completed squares in the exponent: e− y ,Λ y + B k , y = e− w,Λw 1 e4 E k ,k What is the vector b? What is the matrix E ? (iv) How is E related to the symmetric matrix A? (v) Show that 1 J = π n/2 c e 4 E k,k and express c and E in terms of A only. (vi) Does your result agree with the solution to problem 3. when n = 1? 2 ...
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This note was uploaded on 11/09/2011 for the course MATH 601 taught by Professor Un during the Winter '08 term at Ohio State.

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HW4_602_2011 - Math 602 Homework 4 1 Consider the quadratic form on R3 x2 − 2x2 x2 4x1 x2 8x1 x3 4x2 x3 = F(x 1 2 3 Find the following a an

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