EGR 6013 Homework Set 3, Due 2009-10-05

— David Wagner 2009/09/27 13:32

EGR 6013 Homework Set 3, Due 2009-10-05

§1.14, §1.15 Problem 56.

If A is a (homgenous) quadratic form in x_1, x_2,cdots,x_n , prove that
$A = {1/2} sum{k=1}{n}{ x_k {{partial A}/{partial x_k}} }$ .

§1.14, §1.15 Problem 57.

Construct an orthonormal modal matrix Q corresponding to the matrix
A = delim{[}{
matrix{3}{3}{ 1 0 0 0 3 {-1} 0 {-1} 3 }
}{]} .

[A]{x} - λ{x} = {0} ⇒ ([A] - lambda[I]){x} = 0

= $delim{[}{matrix{3}{3}{ (1-lambda) 0 0 0 (3-lambda) {-1} 0 {-1} (3-lambda) } }{]} delim{lbrace}{matrix{3}{1}{ x_1 x_2 x_3 } }{rbrace} =delim{lbrace}{matrix{3}{1}{ 0 0 0 } }{rbrace}$

|[A]-λ[I]| = 0

= (1-lambda)(3-lambda)(3-lambda)-(1-lambda) = (3 -4lambda +lambda^2)(3-lambda)-1+lambda) = (9-3lambda -12lambda +4lambda^2 +3lambda^2 -lambda^3)-1+lambda) = ${-lambda^3 +7 lambda^2 -14 lambda +8}$ = {-(x -1)(x -2)(x -4)}

⇒ lambda_1 = 1, lambda_2 = 2, lambda_3 = 4

√§1.14, §1.15 Problem 60. (a)

Prove that the product of two orthogonal matrices is also an orthogonal matrix.

Given two conformal matrices [A] and [B], both of which are orthogonal, prove that their product [A][B] is also orthogonal.

First, by the definition of an orthogonal matrix, [A] and [B] are square and invertible, with their inverses equal to their transposes. Also, as proven previously, the definition of an orthogonal matrix [Q]^T=[Q]^-1 is equivalent to [Q]^T[Q]=[I].

[A]^T[A] = [I]
[B]^T[A]^T[A][B] = [B]^T[I][B]
([A][B])^T[A][B] = [B]^T[B]
([A][B])^T[A][B] = [I]

Since the product of its transpose and [A][B] is the identity, [A][B] is orthogonal.

√§1.14, §1.15 Problem 60. (b)

Prove that the inverse of an orthogonal matrix is also an orthogonal matrix.

Given an orthogonal invertible (nonsingular) matrix [A], show that [A]^-1 is also orthogonal.

By the definition of transpose,

([A]^T)^T = [A].
([A]^T)^T[A]^T = [A][A]^T.
([A]^-1)^T[A]^-1 = [I].

This last shows that the inverse of [A] is orthogonal as it is equivalent to the definition of orthogonal as proven previously.

§1.18 Problem 69.

Show that if the first two columns of an orthogonal matrix Q comprise the elements of two characteristic vectors of a real symmetric matrix A, then Q^-1 A Q is of the form
$delim{[}{ matrix{5}{5}{ lambda_1 0 0 cdots 0 0 lambda_2 0 cdots 0 0 0 {alpha_{33}} cdots {alpha_{3n}} cdots cdots cdots cdots cdots 0 0 {alpha_{3n}} cdots {alpha_{nn}} } }{]}$ ,
where λ₁ and λ₁ are the characteristic numbers corresponding respectively to the two characteristic vectors.

√§1.19 Problem 72.

Determine whether the real form

$A = {x_1}^2 + 2 {x_2}^2 + {x_3}^2 -2 {x_1} x_2 + 2 x_2 x_3$

is positive definite, by examining the characteristic numbers of the associated matrix.

[A] = delim{[}{matrix{3}{3}{
1 {-1} 0
{-1} 2 1
0 1 1
} }{]}

[A]{x} - λ{x} = 0 = $delim{[}{matrix{3}{3}{ (1-lambda) {-1} 0 {-1} (2-lambda) 1 0 1 (1-lambda) } }{]} delim{lbrace}{matrix{3}{1}{ x_1 x_2 x_3 } }{rbrace}$

|A - λ[I]| = 0 = (1-lambda)(2-lambda)(1-lambda)-(1-lambda)(1-lambda) = (2-lambda)(1-lambda)^2-(1-lambda)^2 = (2-lambda-1)(1-lambda)^2 = (1-lambda)(1-lambda)^2 = (1-lambda)^3

The characteristic number is one of multiplicity three. This number is (or these numbers are all) positive, and since [A] is also symmetric, it is positive definite.

§1.19 Problem 73.

Determine if a real change in the variables which reduces the forms

$A = 3 {x_1}^2 - 2 x_1 x_2 + 3 {x_2}^2$ , $B = 2 {x_1}^2 + 2 {x_2}^2$

simultaneously to the canonical forms

$A = lambda_1 {alpha_1}^2 + lambda_2 {alpha_2}^2$ , $B = {alpha_1}^2 + {alpha_2}^2$

by using the methods of §1.19.

√§1.20 Problem 75.

Determine whether the matrix A of problem 74 is positive definite.

A = delim{[}{
matrix{4}{4}{
2 1 {-1} 0
1 3 4 2
{-1} 4 1 2
0 2 2 1
}
}{]}

This matrix is real and symmetric.

OK.
$beta_2 = |matrix{2}{2}{2 1 1 3}| + |matrix{2}{2}{3 4 4 1}| + |matrix{2}{2}{1 2 2 1}| >0$ OK.
$beta_3 = |matrix{3}{3}{2 1 {-1} 1 3 4 {-1} 4 1}| + |matrix{3}{3}{3 4 2 4 1 2 2 2 1}|$ = 6-4-4 -3-32-1 +3+16+16-4-4+3 = -2 -36 +35-5 < Uh-oh.

This means [A] is not positive definite.

OK.
OK.
Uh-oh.

This is an easier way to show [A] is not positive definite.

§1.20 Problem 78.

Prove that if P is a nonsingular real matrix then A = P^TP is positive definite.

Given [P] is conformal with its transpose implies [P] is square with order n.
[A] also has order n.
Since [P] is nonsingular, it has rank n and det(P) = det(P^T) ≠ 0.

From the definition of the matrix product $[A] = [a_{ij}] = [sum{k=1}{n}{ b_{ik} c_{kj} }]$ , where C = P = $c_{kj} = p_{kj}$ , and B = P^T = $b_{ik}=p_{ki}$ ,

P^TP = $[p_{ki}][p_{kj}] = [sum{k=1}{n}{ p_{ki} p_{kj} }] = [a_{ij}]$

From this, [A] must be symmetric, since $[a_{ji}] = [sum{k=1}{n}{ p_{kj} p_{ki} }] = [sum{k=1}{n}{ p_{ki} p_{kj} }] = [a_{ij}]$ .

The elements of the main diagonal of [A] are all positive, since $[a_{ii}] = [sum{k=1}{n}{ p_{ki} p_{ki} }] = [sum{k=1}{n}{ p_{ki}^2 }]$ .

Now if I could show the rest of the discriminants are positive, this proof would be done.

§1.21 Problem 80.

A geometrical vector ℵ is represented by the numerical vector x = {1,1,1} in terms of components along unit vectors i₁, i₂, and i₃ coinciding with the axes of a rectangular x₁x₂x₃ coordinate system. If new axes are chosen in such a way that the new unit vectors are related to the original ones by the equations
${i_1}prime = {{i_1 + i_2}/{sqrt{2}}}$ , ${i_2}prime = {{i_1 - i_2}/{sqrt{2}}}$ , ${i_3}prime = i_3$ ,
determine the representation x' of ℵ in terms of components of ℵ along the new axes. Show also that the new coordinate system is also rectangular.

§1.21 Problem 81.

A numerical vector y, representing Y, is related to the numerical vector x of Problem 80 by the equation y = Ax, where
A = delim{[}{
matrix{3}{3}{
1 1 {1~}
0 1 {1~}
0 0 {1~}
}
}{]} .
Determine the components of the representation y' in the new system, first, by determining the components of y and transforming them directly, and second, by using equation (199) in connection with the result of Problem 80.

§1.21 Problem 83. (a)

Show that an orthogonal matrix of order two is necessarily of one of the following two types:
$Q^(+) = delim{[}{ matrix{2}{2}{ {cos alpha} {sin alpha} {-sin alpha} {cos alpha} } }{]}$ , $Q^({- ~}) = delim{[}{ matrix{2}{2}{ {cos alpha} {sin alpha} {sin alpha} {-cos alpha} } }{]}$ .
[Notice that |Q⁽⁺⁾| = +1, and |Q^(-)| = -1.]

[Q] = delim{[}{matrix{2}{2}{a b c d} }{]} ⇒ |Q| = ad - bc

[Q]^T[Q] = [I] = delim{[}{matrix{2}{2}{a c b d} }{]} delim{[}{matrix{2}{2}{a b c d} }{]} = $delim{[}{matrix{2}{2}{(a^2+c^2) (ab+cd) (ab+cd) (b^2+d^2)} }{]}$ = delim{[}{matrix{2}{2}{1 0 0 1} }{]}

⇒ a^2+c^2 = 1 , b^2+d^2 = 1 , and ab + cd = 0

$Q^(+) = delim{[}{ matrix{2}{2}{ {x/sqrt{x^2+y^2}} {y/sqrt{x^2+y^2}} {-y/sqrt{x^2+y^2}} {x/sqrt{x^2+y^2}} } }{]}$

§1.21 Problem 83. (b)

If x and x' are considered as two distinct vectors referred to the same axes, and are related by the equation x = Qx', verify that x is rotated into x' through the angle α by a positive (counterclockwise) rotation if Q = Q⁽⁺⁾.

§1.21 Problem 83. (c)

If x and x' are considered as comprising the components of representations of the same geometrical vector, referred to original and rotated axes, respectively, verify that the coordinate transformation x = Q⁽⁺⁾x' corresponds to a negative rotation of the original axes, through the angle α.

§1.21 Problem 83. (d)

If Q = Q^(-) in parts (b) and ©, verify that the transformations then each involve a reversed rotation combined with a suitable reflection.