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 full mathematical viewpoint, this handout contains the important details. We took some pains to write a readable account of wedge products at the cost of straying way beyond the needs of random matrix theory. The steps to understanding how to use wedge products for practical calculations are very straightforward. 1. Learn how to wedge two quantities together. This is usually understood instantly. 2. Recognize that the wedge product is a formalism for computing determinants. This is understood instantly as well, and at this point many readers wonder what else is needed. 3. Practice wedging the order n or mn or some such entries of a matrix together. This requires mastering three good examples. 4. Learn the mathematical interpretation of the wedge product of such quantities as the lower triangular part of QdQ and how this relates to the Jacobian for QR or the symmetric eigendecomposition. We provide a thorough explanation.