18 . 325 : Finite Random Matrix Theory
نویسنده
چکیده
In this section, we concern ourselves with the differentiation of matrices. Differentiating matrix and vector functions is not significantly harder than differentiating scalar functions except that we need notation to keep track of the variables. We titled this section “matrix and vector” differentiation, but of course it is the function that we differentiate. The matrix or vector is just a notational package for the scalar functions involved. In the end, a derivative is nothing more than the “linearization” of all the involved functions. We find it useful to think of this linearization both symbolically (for manipulative purposes) as well as numerically (in the sense of small numerical perturbations). The differential notation idea captures these viewpoints very well. We begin with the familiar product rule for scalars,
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18 . 325 : Finite Random Matrix
There is a wedge product notation that can facilitate the computation of matrix Jacobians. In point of fact, it is never needed at all. The reader who can follow the derivation of the Jacobian in Handout #2 is well equipped to never use wedge products. The notation also expresses the concept of volume on curved surfaces. For advanced readers who truly wish to understand exterior products from a...
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