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:tr(''A'') = ''A''1,1 + ''A''2,2 + ... + ''A''''n'',''n''. where ''A''ij represents the (''i'',''j'')'th element of A. The use of the term ''trace'' arises from the German term '''' (''). PROPERTIES The trace is a Linear Map . That is, :tr(''A + B'') = tr(''A'') + tr(''B'') :tr(''rA'') = ''r'' tr(''A'') for all square matrices ''A'' and ''B'', and all Scalar s ''r''. Since the principal diagonal is not moved on Transposition , a matrix and its transpose have the same trace: :tr(''A'') = tr(''A''T). If ''A'' is an ''n''×''m'' matrix and ''B'' is an ''m''×''n'' matrix, then :tr(''AB'') = tr(''BA''). Note here that ''AB'' is an ''n''×''n'' matrix, while ''BA'' is an ''m''×''m'' matrix. We prove this by invoking the definition of Matrix Multiplication : :, where ''A'' and ''B'' are ''n''x''n'' matrices. Using this fact, we can deduce that the trace of a product of square matrices is equal to the trace of any cyclic permutation of the product, a fact known as the ''cyclic property'' of the trace. For example, with three square matrices ''A'', ''B'', and ''C'', :tr(''ABC'') = tr(''CAB'') = tr(''BCA''). More generally, the same is true if the matrices are not assumed to be square, but are so shaped such that all of these products exist. If ''A'', ''B'', and ''C'' are Square Matrices of the same dimension and are Symmetric , then the traces of their products are invariant not only under cyclic permutations but under all permutations, i.e., :tr(''ABC'') = tr(''CAB'') = tr(''BCA'') = tr(''BAC'') = tr(''CBA'') = tr(''ACB''). The trace is Similarity-invariant , which means that ''A'' and ''P''−1''AP'' (''P'' invertible) have the same trace, though there exist matrices which have the same trace but are not similar. This can be verified using the cyclic property above: : tr(''P''−1''AP'') = tr(''PP''−1''A'') = tr(''A'')
Eigenvalue relationships If ''A'' is a square ''n''-by-''n'' matrix with Real or Complex entries and if λ1,...,λ''n'' are the (complex) Eigenvalue s of ''A'' (listed according to their Algebraic Multiplicities ), then :tr(''A'') = ∑ λ''i''. This follows from the fact that ''A'' is always similar to its Jordan Form , an upper Triangular Matrix having λ1,...,λ''n'' on the main diagonal. From the connection between the trace and the eigenvalues, one can derive a connection between the trace function, the Matrix Exponential function, and the Determinant : :det(exp(''A'')) = exp(tr(''A'')). The trace also prominently appears in Jacobi's Formula for the Derivative of the determinant (see under Determinant ). OTHER IDEAS AND APPLICATIONS If one imagines that the matrix ''A'' describes a water flow, in the sense that for every x in '''R'''''n'', the vector ''A''x 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 '''R'''''n'', 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 Representation s. Given two representations ''A''(''x'') and ''B''(''x''), they are equivalent if tr ''A''(''x'') = tr ''B''(''x''). A matrix whose trace is Zero is said to be ''traceless'' or ''tracefree''. INNER PRODUCT For an ''m''-by-''n'' matrix ''A'' with complex (or real) entries, we have
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 ''n''2. GENERALIZATION The concept of trace of a matrix is generalised to the Trace Class of bounded Linear Operator s on Hilbert Space s. Partial Trace is another generalization of the trace. SEE ALSO |