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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.
 Winter '08
 un
 Linear Algebra, Algebra, Vectors

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