Ans 5 - v1

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 Mm• 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 Mm• 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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern