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B .. C

^1

e (10.29)

B.C

@A

0

The desired operator Q can be found with a Householder transformation. For a given

unit vector n the Householder transformation is de ned by:

^

Q I 2^nT :

n^ (10.30)

;

Problem b: Show that the Householder transformation can be written as Q = I 2P, ;

where P is the operator for projection along n.

^

Problem c: It follows from (10.3) that any vector v can be decomposed in a component

along n and a perpendicular component: v = n (^ v) + v? . Show that after the

^ ^n

Householder transformation the vector is given by:

Qv = n (^ v) + v?

n (10.31)

;^

Problem d: Convince yourself that the Householder transformation of v is correctly

shown in gure (10.4).

Problem e: Use equation (10.31) to show that Q does not change the 0length of a vector.

Use this result to show that a011 in equation (10.28) is given by a11 =

juj.

10.3. THE HOUSEHOLDER TRANSFORMATION 111

^^

n(n . v )

^

n

v

v

Qv

Figure 10.4: Geometrical interpretation of the Householder transformation.

With (10.28) this means that the Householder transformation should satisfy

Qu = ^1 :e (10.32)

juj

Our goal is now to nd a unit vector n such that this expression is satis ed.

^

Problem f: Use (10.30) to show if Q satis es the requirement (10.32) that n must satisfy

^

the following equation:

2^ (^ u) = u ^1

nn ^ ^ e (10.33)

;

in this expression u is the unit vector in the direction u.

^

Problem g: Equation (10.33) implies that n is directed in the direction of the vector

^

u ^1 , therefore n can be written as n =C (^p ^1 ), with C an undetermined con-

^e ^ ^ ue

; ;

stant. Show that (10.32) implies that C = 1= 2 (1 (^ ^1 )). Also show that this

ue

;

value of C indeed leads to a vector n that is of unit length.

^

This value of C implies that the unit vector n to be used in the Householder transformation

^

(10.30) is given by

n = p2 (1u (^1 ^ )) :

^^ e

^ (10.34)

;

u e1

;

To see how the Householder transformation can be used to render the matrix elements be-

low the diagonal equal to zero apply the transformation Q to the linear equation Ax = y.

Problem h: Show that this leads to a new system of equations given by

0 1

a012 a01N

B 0 a022 a02N C

juj

B

B .. .. . . .. C x = Qy :

C (10.35)

B. . . .C

@ A

0 aN2 aNN

0 0

CHAPTER 10. LINEAR ALGEBRA

112

A second Householder transformation can now be applied to render all elements in the

second column below the diagonal element a022 equal to zero. In this way, all the columns of

A can successively be swiped. Note that in order to apply the Householder transformation

one only needs to compute the expressions (10.34) and (10.30) one needs to carry out a

matrix multiplication. These operations can be carried out e ciently on computers.

10.4 The Coriolis force and Centrifugal force

As an example of working with the cross-product of vectors we consider the inertia forces

that occur in the mechanics of rotating coordinate systems. This is of great importance in

the earth sciences, because the rotation of the earth plays a crucial role in the motion of

wind and currents in the atmosphere and in the ocean. In addition, the earth's rotation

is essential for the generation of the magnetic eld of the earth in the outer core.

In order to describe the motion of a particle in a rotating coordinate system we need to

characterize the rotation somehow. This can be achieved by introducing a vector that

is aligned with the rotation axis and whose length is given by rate of rotation expressed

in radians per seconds.

Problem a: Compute the direction of and the length = for the earth's rotation.

j j

â„¦

q

q

=

. q

b

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