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Ans 5 - v1 - G ENERAL ] ARTICLE O n Trace Zero Matrices V S...

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GENERAL ] ARTICLE On Trace Zero Matrices V S Sunder V S Sunder is at the Institute of Mathematical Sciences, Chennai. In this note, we shall try to present an elemen- tary proof of a couple of closely related results which have both proved quite useful, and also indicate possible generalisations. The results we have in mind are the following facts: (a) A complex n • n matrix A has trace 0 if and only if it is expressible in the form A = PQ - QP for some P, Q. (b) The numerical range of a bounded linear op- erator T on a complex Hilbert space 7/, which is defined by W(T) = {(Tx, x) : x e n, IIxll = 1}, is a convex set 1 This result is known - see [1] - as the Toeplitz-Hausdorff theo- rem; in the statement of the iheo- rem, we use standard set-theo- retical notation, whereby x e 5 means that x is an element of the set 5. Keywords Inner product, commutator, con- vex set, Hilbert space, bounded linear operator, numerical range. We shall attempt to make the treatment easy- paced and self-contained. (In particular, all the terms in 'facts (a) and (b)' above will be~ de- scribed in detail.) So we shall begin with an introductory section pertaining to matrices and inner product spaces. This introductory section may be safely skipped by those readers who may be already acquainted with these topics; it is in- tended for those readers who have been denied the pleasure of these acquaintances. Matrices and Inner-product Spaces An m • n matrix is a rectangular array of numbers of the form A= a21 a22 "" a2n . (1) 9 . ... * aml am2 "'" amn 14 RESONANCE [ June 2002
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GENERAL J ARTICLE We shall sometimes simply write A = ((a~j)) as short- hand for the above equation and refer to a~ as the entry in the i-th row and j-th column of the matrix A. The matrix A is said to be a complex m x n matrix if (as in (1)) A is a matrix with m rows and n columns all of whose entries a~j are complex numbers. In symbols, we shall express the last sentence as A Mm• r aii C for all 1 _ i, j < n. (Clearly, we may similarly talk about the sets Mm• and M,~x~(Z) of m • n real or integral matrices, respecti- vely; 2 but we shall restrict ourselves henceforth to com- plex matrices.) The collection M,~xn(C) has a natural structure of a complex vector space in the sense that if A = ((aij)), B = ((b~j)) M,~xn(C) and A C, we may define the linear combination AA + B to be the matrix with (i,j)-th entry given by Aa~j + b~j. (The 'zero' of this vector space is the m • n matrix all of whose entries are 0; this 'zero matrix' Will be denoted simply by 0.) Given two matrices whose 'sizes are suitably compati- ble', they may be multiplied. The product AB of two matrices A and B is defined only if there are integers m,n,p such that A = ((aik)) e Mmx,~, B = ((bkj)) M~• in that case AB is defined as the matrix ( (c~j) ) given by n ci~ = E aikbkj. (2) k=l Unlike the case of usual numbers, matrix-multiplication is not 'commutative'. For instance, if we set o1) 0) 1 ' 0 0 ' then it may be seen that AB -7/= BA.
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Ans 5 - v1 - G ENERAL ] ARTICLE O n Trace Zero Matrices V S...

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