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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 ToeplitzHausdorff theo
rem; in the statement of the iheo
rem, we use standard settheo
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 selfcontained. (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 Innerproduct 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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View Full DocumentGENERAL
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 ith row and jth 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, matrixmultiplication
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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 Fall '08
 Chandrasekara

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