Purdue University Numerical Linear Algebra Group

January 26 (part 1) and February 2 (part 2), 2015 at Purdue University.

Talk given:

Introduction to the Kronecker Product

Part 1

Definitions, examples, basic properties. In particular, how eigen-information of A \kron B is exactly determined by eigen-information of A and of B. We used a Kronecker product perspective to show an easier way of studying the Poisson matrix that caused so many of us so much pain in CS515.

Part 2

Last time we covered some multiplicative and eigen properties of Kronecker products. We will build on those with some new properties, then look at some neat applications like using the Kronecker product to (1) introduce and solve some matrix equations like the Sylvester and Lyapunov equations, e.g. AX - XA = C; (2) speed up matrix operations; and (3) provide data-sparse representations and approximations of matrices, including a ‘nearest Kronecker-product’ theorem similar to a theorem that says the SVD gives the best low-rank approximation of a matrix.