trace (矩阵的迹) 的性质

trace (矩阵的迹) 的性质

性质1

tr(Am×nBn×m)=tr(BA) t r ( A m × n B n × m ) = t r ( B A )

证明

tr(AB) t r ( A B )
=∑i=1m(AB)ii = ∑ i = 1 m ( A B ) i i
=∑i=1m∑j=1naijbji = ∑ i = 1 m ∑ j = 1 n a i j b j i
=∑j=1n∑i=1mbjiaij = ∑ j = 1 n ∑ i = 1 m b j i a i j
=∑j=1n(BA)jj = ∑ j = 1 n ( B A ) j j
=tr(BA) = t r ( B A )

性质2

tr(Am×pBp×nCn×m) t r ( A m × p B p × n C n × m )
=∑i=1m(ABC)ii = ∑ i = 1 m ( A B C ) i i
=∑i=1mai,:(BC):,i = ∑ i = 1 m a i , : ( B C ) : , i
=∑i=1mai,:Bc:,i = ∑ i = 1 m a i , : B c : , i