HW4_602_2012

# HW4_602_2012 - Math 602 Homework 4 1. Consider the...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 e-ax +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 = . - 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,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: e-y,y+Bk,y = e-w,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? 1 2 ...
View Full Document

## 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.

Ask a homework question - tutors are online