Trace (linear Algebra) - Example

Example

Let T be a linear operator represented by the matrix

\begin{bmatrix}-2&2&-4\\ -1& 1& 3\\ 2 &0 &-1\end{bmatrix}.

Then tr(T) = −2 + 1 − 1 = −2.

Assume the characteristic of the field is zero. Then the trace of the identity matrix is the dimension of the space; this leads to generalizations of dimension using trace. The trace of a projection (i.e., P2 = P) is the rank of the projection. The trace of a nilpotent matrix is zero. The product of a symmetric matrix and a skew-symmetric matrix has zero trace.

More generally, if f(x) = (x − λ1)d1···(x − λk)dk is the characteristic polynomial of a matrix A, then

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