On bounds for scaled projections and pseudoinverses

On Bounds for Scaled Projectlons and Pseudoinverses Dianne

P. O’Leary*

Computer

Science

and Institute

Department Computer

fm Advanced

University of Maryland College Park, Maryland

Studies

20742

Submitted by G. W. Stewart

ABSTRACT Let X be a matrix of full column rank, and let D be a positive definite diagonal matrix. In a recent paper, Stewart considered the weighted pseudoinverse XL = (X’DX)-‘X’D and the associated oblique projection I’o = XXA, and gave bounds, independent of D, for the norms of these matrices. In this note, we answer a question he raised by showing that the bounds are computable.

Let X be a matrix of full column rank, and let D be a positive definite diagonal matrix. In a recent aper, Stewart [l] considered the weighted P XTD and the associated oblique projection pseudoinverse X 2, = (XTDX)PD = XXA. He proved two results. The first is that the spectral norms of these

matrices

are bounded

independently

sup

of D as

IlPDll G P-’

DE3+

and sup

Il$Jll GP-lllx+ll~

DEL.3,

*This work was supported 87-0188.

by the Air Force Of&e

LINEAR ALGEBRA AND ITS APPLZCATZONS 0 Elsevier Science Publishing Co., Inc., 1990 655 Avenue of the Americas, New York, NY 10010

of Scientific

132:115-117

Research under Grant

(1990)

115

0024-3795/90/$3.50

116

DIANNE

P. O’LEARY

where

with

His second result 3?(X), then

X=

{xE

9

= {y : 30

.qX):J(xl(

U, denotes

In this note,

is that if the columns

X“Dy

of U form

= 0).

(3)

an orthonormal

basis

mininf+(U,),

any s&matrix we answer

(2)

E _9+ such that

p< where

= l},

formed

a question

for

(4)

from a nonempty he raised

by showing

set of rows

of U.

that

p = mininf+(U,). Since X and W depend only on the range can replace X in (2) and (3) by U. Thus,

Let the sign of a scalar

t be defined 1 0 i -1

sg(t)=

of X and not on its entries,

we

by if if if

t > 0, t=O, t
and let the sign of a vector z be denoted by sg(z) and defined componentwise. Then ?Y has the property that for any vector Q E WV, every vector y with sg( y) = sg( 9) is also an element of EV. This is verified by letting D be the nonnegative diagonal matrix such that U’Dtj = 0. Then UTDSy = 0, where S is the diagonal matrix with

sii =

9i

i 1

/

Yi

if

yi # 0,

if

yi=O.

SCALED

117

PROJECTIONS

Now,

In the inner infimum, for every choice of (Y there is a set of rows of Ua that agree in sign with y and a set that disagree. The set of rows that disagree in sign must be nonempty; zero, which contradicts

otherwise y = Ua E V, and the infimum would (1). Let the set of those that disagree be denoted

the subscript 1. For this choice of (Y, the best y equals Ua in agree in sign and has elements zero or arbitrarily close to zero rows. The resulting value of lly - Uall is no less than Il(Ucu>,ll this value is bounded below by the smallest singular value of have shown that pa and combining

this with

be by

all rows that in the other = llU,cyIl, and U,. Thus we

mininf+(U,),

Stewart’s

result

(4) establishes

the equality.

REFERENCES 1

G. W. Stewart, On Scaled Projections 112 (1989) 189-194. Received 3 March 1989; final mmuxript

and Pseudoinverses,

accepted 9 March 1989

Linear Algebra Appl.,