Redirected from Trace of a matrix
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.
The trace is a linear map in the sense that
A matrix and its transpose have the same trace:
If A is an n×m matrix and B is an m×n matrix, then
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,
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
From the connection between the trace and the eigenvalues, one can derive a connection between the trace function, the exponential function, and the determinant:
For an m-by-n matrix A with complex (or real) entries, we have
If m=n then the norm induced by the above inner product is called the Frobenius norm[?] of a square matricx. Indeed it is simply the Euclidean norm[?] if the matrix is considered as a vector of length n2.
The concept of trace of a matrix is generalised to the trace class of bounded linear operators on Hilbert spaces.
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