44 Matrix trace
The trace of a square
44.1 Properties of trace
For any
and ,An immediate consequence of the above is that for any
, , and ,For
and any scalars and ,In fact, it can be shown that trace is the only function
that satisfies the following three properties: for any and scalars , : . . .
For a square matrix, trace is equal to the sum of the eigenvalues, i.e.,
For positive semidefinite matrices (of the same size), trace is sub-multiplicative, i.e.,
Thus,
.The above can be generalized to arbitrary powers. For positive semidefinite matrices (of the same size), and for any positive integer
: andAnother generalization is the following. Let
be positive semidefinite matrices (of the same size) and are positive numbers such that Let Then, andThe next result removes the restriction to positive semidefinite matrices (at the cost of working with absolute values). Let
be arbitrary matrices and are positive numbers such that Let Then, for any integer , and
Notes
See Yang and Feng (2002) and Shebrawai and Albadawani (2012) for generalizations of the above inequalities.