for absence of constraints encompassing j and l; Eq. (4.2) was again followed, coupled with the distributive property of multiplication of scalars. The trace of ( BC) A abides to
based on its definition; the right‐hand side of Eq. (4.98) is identical to the right‐hand side of Eq. (4.95), so one readily concludes that
(4.99)
– whereas combination with Eq. (4.57) leads finally to
Remember that BC is an (n × m) matrix and A is an (m × n) matrix, so ( BC) A = BCA is an (n × n) matrix – and thus distinct from the (m × m) matrix ( AB) C = ABC as outlined above, due to the product of matrices not being commutative; nevertheless, the traces of BCA and ABC are the same. A similar derivation would prove that
(4.101)
where (p × n) matrix CA is compatible with (n × p) matrix B, thus yielding a (p × p) matrix CAB that possesses a trace for being square – and equal to that of BCA, despite CAB being distinct from (n × n) matrix BCA . Note that the next similar move of swapping the first factor (i.e. C) to the last position without modifying the sequence of the other two (i.e. AB) would transform tr{ CAB } to tr{ ABC } again – so tr{ CAB } = tr{ ABC } would close the cycle with Eqs. (4.100) and (4.101).
A particular situation covered by Eq. (4.100) pertains to an (n × m) matrix A and an (m × n) matrix C, together with Im playing the role of matrix B; the products ABC and BCA in Eq. (4.100) look like
(4.102)
which degenerates to
(4.103)
at the expense of Eqs. (4.61) and (4.64). Therefore, the trace of the product of two matrices remains unchanged when the said matrices switch positions (should that be compatible with multiplication).
4.4 Transposal of Matrices
Recall again matrix A, as defined by Eqs. (4.1) and (4.2); if generic element ai,j, initially located in the ith row and jth column, were swapped with element aj,i, initially located in the jth row and ith column, then a transpose matrix would result – given by
in more condensed form, Eq. (4.104) reads
so AT will be an (n × m) matrix. The order of a square matrix is obviously not changed upon transposal – neither do the diagonal elements (characterized by i = j), irrespective of its order; hence, one finds that
(4.106)
for a square A . Furthermore, if the elements symmetrically located with regard to the main diagonal are identical, then
a matrix bearing this property is termed symmetric – a concept distinct from that conveyed by Eq. (4.44) that involves two matrices. A particular case of the above statement is the identity matrix – since ai,j≠i = 0 = aj,i; hence,
Application of Eq. (4.105) twice sequentially supports
(4.109)
which may be condensed to
therefore, the inverse of transposal coincides with transposal itself, as composition of the two leaves the original matrix unchanged.
When transposal is combined with addition of matrices, one obtains
(4.111)
from Eqs. (4.2) and (4.3), after direct combination with Eq. (4.105); whereas
