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The trace of a Hermitian matrix is real, because the elements on the diagonal are real. The trace of a permutation matrix is the number of fixed points of the corresponding permutation, because the diagonal term a ii is 1 if the i th point is fixed and 0 otherwise. The trace of a projection matrix is the dimension of the target space.
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [ 1 ] If A is a differentiable map from the real numbers to n × n matrices, then
The trace operator has however a well-behaved right-inverse, which extends a function defined on the boundary to the whole domain. Specifically, for 1 < p < ∞ {\textstyle 1<p<\infty } there exists a bounded, linear trace extension operator [ 3 ]
[a] This means that the function that maps y to f(x) + J(x) ⋅ (y – x) is the best linear approximation of f(y) for all points y close to x. The linear map h → J(x) ⋅ h is known as the derivative or the differential of f at x. When m = n, the Jacobian matrix is square, so its determinant is a well-defined function of x, known as the ...
The covariant derivative of a function ... defined as the -trace of the second fundamental form. Then ... The variation formula computations above define the ...
by and the definition of the trace. It remains to show that this representation of the derivative implies Liouville's formula. Fix x 0 ∈ I. Since the trace of A is assumed to be continuous function on I, it is bounded on every closed and bounded subinterval of I and therefore integrable, hence
Trace formula may refer to: Arthur–Selberg trace formula, also known as invariant trace formula, Jacquet's relative trace formula, simple trace formula, stable trace formula; Grothendieck trace formula, an analogue in algebraic geometry of the Lefschetz fixed-point theorem in algebraic topology, used to express the Hasse–Weil zeta function.
The Laplace–Beltrami operator, when applied to a function, is the trace (tr) of the function's Hessian: = (()) where the trace is taken with respect to the inverse of the metric tensor. The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on tensor fields , by ...