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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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