PROBLEMS AND SOLUTIONS 83

this angle is "/T- (02 "[" 03)/2; and the 0j have sum , whence

COS b 1.2 ]

A similar argument yields the cycled identity

(15) n. 3 Xl sin (. + 0) (0)2cos bz" 2,

while the triangle has the area [al a:l/2. Finally,

max [det (A)I la x a21" {[all(b 1)(n 2 %) + la2l(b2 2)(n

(16) =(area of a). {la11(1 +cos 0) + la2[(a +cos 02)}

(area of ). (perimeter of ).

The proposer’s recent work [1] evaluates this and related maxima.

REFERENCE

[1] L. S. LEW, Optimal accelerometer layouts for signature Oerification, Research Rep. RC 7374, IBM T.J. Watson Research Center, Yorktown Heights, NY 1978.

A Constrained Matrix Optimization Problem

Problem 81-4, by H. WOLKOWICZ (University of Alberta).Given a real symmetric n x n matrix B and three subspaces L1, La and L3 of R n,

determine the (unique) real symmetric n x n matrix A which is closest to B in theEuclidean norm (Hilbert-Schmidt norm) and which is negative semi-definite (nsd) onL1, positive semi-definite (psd) on La and 0 on L3.

Editorial note. A solution of this problem is given in 5 of the proposer’s paperSome applications of optimization in matrix theory, to appear in Linear Algebra andits Applications.

Principal Value of an Integral

Problem 81-5", by H. E. FETTIS (Mountain View, California).It is known that

cos h0 dO 7r sinPV

cos 4 -cos 0 sin bwhen h is an integer (PV denotes the Cauchy principal value of the integral). Evaluatethe integral for nonintegral h.

Solution by JAMES A. BOA (SUNY at Buffalo).

We have 2I PVcos h0 dO

PVe ix dO

cos 4 -cos 0 cos 4 -cos 0

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-2iz(z) :=

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