Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 3 / 10H / a

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- Solution a

Solution

The matrix trace is the sum of diagonal entries,

tr A = \sum_{i = 1}^{n} A_{ii}

. For any two square matrices,

tr (BC) = \sum_{i, j} B_{ij} C_{ji} = \sum_{j, i} C_{ji} B_{ij} = tr (CB) .

Apply this cyclic property of the trace with

B = T

and

C = A T^{- 1}

:

tr (T A T^{- 1}) = tr (A T^{- 1} T) = tr A .

Thus matrix similarity preserves the matrix trace; no assumption that either matrix is diagonalizable is needed.

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