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Trace (linear algebra)

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.,

:\mathrm{tr}(A) = a_{11} a_{22} \dots a_{nn}=\sum_{i=1}^{n} a_{i i} \,
where aii represents the entry on the i th row and i th column of A . Equivalently, the trace of a matrix is the sum of its [eigenvalue] s, making it an [invariant] with respect to a [change of basis] . This characterization can be used to define the trace for a linear operator in general. Note that the trace is only defined for a square matrix (i.e. n × n ).

Geometrically, the trace can be interpreted as the infinitesimal change in volume (as the derivative of the [determinant] ), which is made precise in [Jacobi's formula] .

The use of the term trace arises from the German term ( [cognate] with the English spoor ), which, as a function in mathematics, is often abbreviated to "Sp".

Examples
Let T be a linear operator represented by the matrix

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

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., P 2 = 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 is the [characteristic polynomial] of a matrix A , then
:\mathrm{tr}(A) = d_1 \lambda_1 \cdots d_k \lambda_k.\!

If A and B are [positive semi-definite matrices] of the same order then

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