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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 {1 , 2 , 3 }, and a set {1 , 2 , 3 } e e e so that 3 3 2 = x yii F (x) = e i yi
1 1 b) Find an orthogonal matrix U such that = U F (x) = x y c) Find the max and min of F (x) subject to
i=1 3 x2 = 1. i 2 1 2. Calculate the square root R of the matrix A = 1 2 Verify that the matrix R you found is indeed positive semidefinite and that R2 = A. 3. Let a, k be constants, with a > 0. Calculate 2 eax +kx dx
 3 1 2 i yi . by completing the square and using the fact that 4. Consider the integral J   e y 2 dy = .  ex,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= ey,y+Bk,y dy1 dyn
  where 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 b, (ii) By shifting the coordinates y = (y1 , , yn ) by a vectorial amount y =w+b bring the exponential under the integral into a form with completed squares in the exponent: ey,y+Bk,y = ew,w e 4 Ek,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 = n/2 c e 4 Ek,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?
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This note was uploaded on 03/18/2012 for the course MATH 602 taught by Professor Costin during the Spring '12 term at Ohio State.
 Spring '12
 COSTIN
 Vectors

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