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HOUSEHOLDER ORTHONORMAL TRANSFORMATION In the preceding chapter we showed how the elementary Givens orthonormal transformation triangularized a matrix by successfully zeroing out one ele

HOUSEHOLDER ORTHONORMAL

TRANSFORMATION

In the preceding chapter we showed how the elementary Givens orthonormal transformation triangularized a matrix by successfully zeroing out one element

at a time below the diagonal of each column With the Householder orthonormal transformation all the elements below the diagonal of a given column are zeroed out simultaneously with one Householder transformation Specifically, with the first Householder transformation H1, all the elements below the first element in the first column are simultaneously zeroed out, resulting in the form given by (11.1-4) With the second Householder transformation H2 all the elements below the second element of the second column are zeroed out, and so on

The Householder orthonormal transformation requires fewer multiplies and adds than does the Givens transformation in order to obtain the transformed upper triangular matrix of (10.2-8) [103] The Householder transformation, however, does not lend itself to a systolic array parallel-processor type of implementation as did the Givens transformation Hence the Householder may

be preferred when a centralized processor is to be used but not if a custom systolic signal processor is used

12.1 COMPARISON OF HOUSEHOLDER AND GIVENS

TRANSFORMATIONS

Let us initially physically interpret the Householder orthonormal transformation

as transforming the augmented matrix T0 to a new coordinate system, as we did for the Givens transformations For the first orthonormal Householder trans-formation H1 the s rows are unit vectors, designated as hifor the ith row, onto

315

Copyright # 1998 John Wiley & Sons, Inc ISBNs: 0-471-18407-1 (Hardback); 0-471-22419-7 (Electronic)

which the columns of T0are projected The first-row vector h1is chosen to line

up with the vector formed by the first column of T0; designated as t1 Hence, the projection of t1onto this coordinate yields a value for the first coordinate in the transformed space equal to the full length of t1, that is, equal to k t1k The remaining s 1 unit row vectors of Hi, designated as hi, i¼ 2; ; s, are chosen to be orthonormal to h1: As a result they are orthonormal to t1 since h1 lines up with t1 Hence the projections of t1 onto the hi; i¼ 2; ; s, are all zero As a result H1T0 produce a transformed matrix of the form given by (11.1-4) as desired This is in contrast to the simple Givens transformations

G1; G2; ; Gs1 of the left-hand side of (11.1-4), which achieves the same outcome as H1 does but in small steps Here, G1 first projects the column vector t1 of T0onto a row space that is identical to that of the column space of

T0 with the exception of the first two coordinates The first two coordinates, designated in Figure 11.1-2a as the x, y coordinates, are altered in the new coordinate system defined by G1 The first of these two new coordinates, whose direction is defined by the first-row unit vector g1, of G1, is lined up with the vector formed by the first two coordinates of t1, designated as t0; see (11.1-1) and Figure 11.1-2a The second of these new coordinates, whose direction is defined by the second-row unit vector g2 of G1, is orthogonal to g1 and in turn t0, but in the plane defined by the first two coordinates of t1, that is, the x, y plane As a result G1T0 gives rise to the form given by (11.1-2)

The second Givens transformation G2 projects the first-column vectorðt1Þ1

of G1T0 onto a row space identical to that ofðt1Þ1except for the first and third coordinates The first-row unit vectorðg1Þ2 of G2is lined up with the direction defined by the vector formed by the first and third coordinates of G1T0 This vector was designated as t2; see (11.1-16) and Figures 11.1-2b,c From the definition of t2recall also that it is lined up with the first three coordinates of t1 The third-row unit vectorðg3Þ2 is lined up in a direction orthogonal toðg1Þ2 in the two-dimensional space formed by the first and third coordinates of the vectorðt1Þ2; see Figures 11.1-2b,c As a result the projection ofðt1Þ1 onto the row space of G2 gives rise to the form given by (11.1-3) Finally, applying the third Givens transformation G3 yields the desired form of (11.1-4) obtained with one Householder transformations H1: For this example T0 is a 4 3 matrix, that is, s¼ 4: For arbitrary s; s  1 simple Givens transformations

G1; ; Gs1 achieve the same form for the transformed matrix that one Householder transformation H1 does, that is,

H1  Gs1   G2G1 ð12:1-1Þ Elements on the right and left hand sides of (12.1-1) will be identical except the sign of corresponding rows can be different if the unit row transform vectors of these transforms have opposite directions; see Section 13.1

Let us recapitulate what the Givens transformations are doing The first, G1, projects T onto a space whose first coordinate unit vector g (defined by the

first row of G1; see Section 11.1) lines up with the direction of the two-dimensional vector formed by the first two coordinates of t1 The second coordinate of the new space is along the unit vector g2 (defined by the second row of G1) orthogonal to the first coordinate but in the plane defined

by the first two coordinates of t1 The remaining coordinates of the space defined by G1 are unchanged In this transformed space the second coordinate

of t1 is zero; see (11.1-2) This in effect replaces two coordinates with one for the first two coordinates of t1, the other coordinate being zero The next Givens transformation now projects the transformed t1, which is ðt1Þ1, onto the row space of G2 Here the first-row unit vector ðg1Þ2 of G2 is lined up with the vector formed by first and third coordinates of ðt1Þ1, which is designated as t2 in (11.1-16) and Figures 11.1-2b,c Again the second coordinate is orthogonal to the first but in the plane defined by the first and third coordinates of ðt1Þ1 In this new coordinate system the second and the third coordinates of t1 are zero; see (11.1-3) This simple Givens transformation again replaces two coordinates of ðt1Þ1, the first and third, with one, the second being zero The first and second simple Givens transfor-mations together in effect line up the first row of G1G2 with the vector formed by the first three coordinates of t1 This continues until on the

s 1 Givens transformation the unit vector formed by the first row of

Gs1   G2G1 lines up with the vector formed by t1 to produce the form given by (11.1-4) In a similar way, the first Householder transformation

H1 does in one transformation what s 1 simple Givens transformations

do Similarly H2 does in one transformation what s 2 Givens transfor-mations do; and so on

12.2 FIRST HOUSEHOLDER TRANSFORMATION

We now develop in equation form the first Householder transformation H1 A geometric development [80, 102, 109, 114] is used in order to give further insight into the Householder transformation Consider the s ðm0þ 1Þ dimensional augmented matrix T0 whose columns are of dimension s As done previously, we will talk of the columns as s-dimensional vectors in s-dimensional hyperspace Designate the first column as the vector t1 Form

h0 given by

h0 ¼ t1þ k t1k i ð12:1-2Þ

where i is the unit vector along the x axis of this s-dimensional hyperspace, that is, the column space of T0; see Figure 12.1-1 The vector i is given by (4.2-38) for s¼ 3 Physically let us view the first Householder transformation

H1 as a rotation of t1 Then i in Figure 12.1-1 is the direction into which we want to rotate t1 in order to have the transformed t1, designated asðt1Þ , be

given by

ðt1Þ1H ¼ H1t1¼ ½k t1k 0 0    0T ð12:1-3Þ that is, in order for the first column of H1T0 to have the form given by the first column of (11.1-4) In Figure 12.1-1, the vector h0 forms the horizontal axis, which is not the x axis that is along the unit vector i Because H1t1 in Figure 12.1-1 is the mirror image of t1 about this horizontal axis, the Householder transformation is called a ‘‘reflection.’’

Let h be the unit vector along h0; then

h¼ h 0

Let t1c be the component of t1 along the direction h Then

t1c¼ ðtT

Designate t1sas the component of t1 along the direction perpendicular to h but

in the plane formed by t1 and i Then

t1s¼ t1 t1c

¼ t1 ðtThÞh ð12:1-6Þ Figure 12.1-1 Householder reflection transformation H1for three-dimensional space

Then from geometry (see Figure 12.1-1)

H1t1¼ t1c t1s

¼ ðtT

1hÞh  ½t1 ðtT

1hÞh

¼ 2ðtT

1hÞh  t1

¼ 2hðtT

1hÞ  t1

¼ 2hhTt1 t1

¼ 2hh
T I

Hence

By picking h as given by (12.1-2) and (12.1-4), the vector t1is rotated (actually reflected) using (12.1-8) onto i as desired After the reflection of t1 the vector has the same magnitude as it had before the reflection, that is, its magnitude is given by k t1 k Moreover, as desired, the magnitude of the first element of the transformed vector ðt1Þ1H has the same magnitude value as the original vector t1 with the value of the rest of the coordinate elements being zero Specifically

H1t1 ¼ H1

t11

t21

ts1

2 6 6 6

3 7 7

7¼

k t1k 0 0

0

2 6 6 6

3 7 7

7¼ ðt1Þ1H ð12:1-9Þ

where

k t1 k¼ ðjt11j2þ jt11j2þ    þ jts1j2Þ1=2 ð12:1-9aÞ The subscript 1 on H is used to indicate that this is the first House-holder transformation on the matrix T0, with more Householder transforms

to follow in order to zero out the elements below the remaining diagonal elements

In the above we have only talked about applying the s s Householder transformation H1 matrix to the first column of T0 In actuality it is applied to all columns of T0 This is achieved by forming the product H1T0 Doing this yields a matrix of the form given by the right-hand side of (11.1-4) as desired

In this way we have physically reflected the first column of T0 onto the first coordinate of the column space of T0 (the x coordinate) by the use of the Householder transformation H

12.3 SECOND AND HIGHER ORDER HOUSEHOLDER

TRANSFORMATIONS

The second Householder transformation H2 is applied to H1T0 to do a collapsing of the vector formed by the lower s 1 elements of the second column of H1T0 onto the second coordinate Designate asðt0

2Þ1H the ðs  1Þ-dimensional vector formed by the lower s 1 elements of the second column of

H1T0, the elements starting with the diagonal element and including all the elements below it It is to be reflected onto the ðs  1Þ-dimensional vector

j0¼ ½100    0T that physically is the unit vector in the direction of the second coordinate of the second column of the matrix H1T0 When this is done, all the elements below the diagonal of the seond column are made equal to zero The top element is unchanged The Householder transformation needed to do this is

an ðs  1Þ  ðs  1Þ matrix that we designate as H0

2: This transformation H02 operates on the ðs  1Þ  ðm0Þ matrix in the lower right-hand corner of the transformed s ðm0þ 1Þ matrix H1T0

From H20 we now form the Householder transformation matrix H2 given by

H2¼ I1 0

0 H20

g 1

where I1is the 1 1 identity matrix and H2 is now an s s matrix When the orthonormal transformation matrix H2 is applied to H1T0; it transforms all elements below the diagonal elements of the second column to zero and causes the transformed second diagonal element to have a magnitude equal to the magnitude of the (s 1)-dimensional column vector ðt0

2Þ1H before the transformation and leaves the top element of the second column unchanged This procedure is now repeated for the third column of the transformed matrix

H2H1T0 and so on, the ith such transformation being given by

Hi¼ Ii1 0

0 H0i

g i  1

g s  i þ 1 ð12:2-2Þ The last transformation needed to triangularize the augmented matrix T0 so as

to have it in the form given by (11.1-30) is i¼ m0þ 1

It is important to point out that it is not necessary to actually compute the Householder transformation Hi’s given above using (12.1-8) The transformed columns are actually computed using the second form of the H1t1 given by (12.1-7), specifically by

H1t1 ¼ t1þ 2ðt1ThÞh ð12:2-3Þ

In turn the above can be written as

H1t1 ¼ t1þ2ðt

T

1h0Þh0

for obtaining the transformed t1 with h0 given by (12.1-2) For detailed computer Householder transformation algorithms the reader is referred to references 79 to 81 Some of these references also discuss the accuracy of using the Householder transformation for solving the least-squares estimate relative to other methods discussed in this book

Some final comments are worth making relative to the Householder trans-formation In the above we applied m0þ 1 Householder transformations so as to transform the augmented matrix T0 (11.1-25) into an upper triangular matrix form Specifically, applying the orthonormal transformation formed by

F¼ Hm 0 þ1Hm 0   H2H1 ð12:2-5Þ

to (11.1-25), we obtain the transformed matrix T00 given by

T00 ¼ F T0¼ F½T jYðnÞ

¼ ½F T jF YðnÞ

¼

U -0 -0

2 6 6 6 6 4

|ffl{zffl}

m0

j -j -j

Y10

-Y20

-Y30

3 7 7 7 7 5

|ffl{zffl}

1

g m0

g 1

g s  m0 1

ð12:2-6Þ

with Y30 ¼ 0 [The above is the same result as obtained in Section 4.3 using the Gram–Schmidt onthogonalization procedure; see (4.3-60).] The vector Y30 is forced to be zero by the last Householder transformation Hm0 þ1 of (12.2-5) To determine X n;n it is actually not necessary to carry out the last Householder transformation with the result that some computational savings are achieved The last Householder transformation does not affect Y10 and in turn X n;n; see (10.2-16) Making Y30 ¼ 0 forces the residue error eðX n;n) of (4.1-31) to be given byðY0

2Þ2, that is,

eðX n;nÞ ¼ ðY20Þ2 ð12:2-7Þ This property was pointed out previously when (11.1-31) and (11.3-3) were given and before that in Section 4.3; see, for example, (4.3-56) If the last Householder transformation Hm 0 þ1 is not carried out, then Y30 is not zero and

eðX n;nÞ ¼ ðY20Þ2þ k Y30 k2 ð12:2-8Þ

A similar comment applies for the Givens transformation of Chapter 11

¼ 2hh
T I

Hence

By picking h as given by (12.1-2) and (12.1-4), the vector t1is rotated (actually reflected) using (12.1-8) onto i... product H1T0 Doing this yields a matrix of the form given by the right-hand side of (11.1-4) as desired

In this way we have physically reflected the first column