In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e.
- tr(A) = A1,1 + A2,2 + ... + An,n .
If one imagines that the matrix A describes a water flow, in the sense that for every x in Rn, the vector Ax represents the velocity of the water at the location x, then the trace of A can be interpreted as follows: given any region U in Rn, the net flow of water out of U is given by tr(A)· vol(U), where vol(U) is the volume of U. See divergence.
The trace is used to define characters of group representations.
Table of contents |
2 Inner Product 3 Generalization |
The trace is a linear map in the sense that
Properties
A matrix and its transpose have the same trace:
- tr(A) = tr(AT).
- tr(AB) = tr(BA).
- tr(ABC) = tr(CAB) = tr(BCA).
If A and B are similar, i.e. if there exists an invertible matrix X such that A = X-1BX, then by the cyclic property,
- tr(A) = tr(B).
There exist matrices which have the same trace but are not similar.
If A is a square n-by-n matrix with real or complex entries and if λ1,...,λn are the (complex) eigenvalues of A (listed according to their algebraic multiplicities), then
- tr(A) = ∑ λi.
From the connection between the trace and the eigenvalues, one can derive a connection between the trace function, the exponential function, and the determinant:
- det(exp(A)) = exp(tr(A)).
For an m-by-n matrix A with complex (or real) entries, we have
Inner Product
with equality only if A = 0.
The assignment
yields an inner product on the space of all complex (or real) m-by-n matrices.
If m=n then the norm induced by the above inner product is called the Frobenius norm of a square matrix. Indeed it is simply the Euclidean norm if the matrix is considered as a vector of length n2.
Generalization
The concept of trace of a matrix is generalised to the trace class of bounded linear operators on Hilbert spaces.