Advanced Linear Algebra - Advanced Linear Algebra, Lecture 7.2: Nonstandard inner products and Gram matrices

Advanced Linear Algebra, Lecture 7.2: Nonstandard inner products and Gram matrices As we have seen, the matrix A*A has a number of nice properties, such as having the same rank and nullity of A itself, being positive nonnegative (i.e., positive semi-definite), and positive if A has full column rank. The real reason behind this is because it is a "matrix of inner products": the (i,j)-entry of the is the inner product (e_i,e_j). This is called a Gram matrix, which defined more generally for vectors x_1,...,x_m in a general inner product space. It is easy to show why the Gram matrix G of a linearly independent set of vectors is positive. But the converse is also true, and this can constructed by defining the "nonstandard inner product" of x and y to be (x,Gy). This also leads to the "generalized Rayleigh quotient, of R_{M,H}=(x,Hx)/(x,Mx), which is actually a standard Rayleigh quotient with respect to a nonstandard inner product. We conclude with a curious result by Schur, which says that the entry-wise product of two positive matrices is also positive, and this follows by considering the resulting matrix as a Gram matrix in a tensor product of vectors spaces. Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html