MATH 1005H Lecture 6: linearalgebra (1)
5 Oct 2015
School
Department
Course
Professor
Document Summary
Below are a few basic properties of matrices: matrix multiplication is associative: (ab)c = a(bc, matrix multiplication is distributive: a(b + c) = ab + ac, matrix multiplication is not commutative in general, that is ab (cid:54)= ba. For example, if a rm n and b rn q, the matrix product ba does not exist. The transpose of a matrix a rm n, is written as a(cid:62) rn m where the entries of the matrix are given by: (a(cid:62))ij = aji (2. 1) Properties: transpose of a scalar is a scalar a(cid:62) = a, (a(cid:62))(cid:62) = a, (ab)(cid:62) = b(cid:62)a(cid:62, (a + b)(cid:62) = a(cid:62) + b(cid:62) The trace of a square matrix a rn n is written as tr(a) and is just the sum of the diagonal elements: The trace of a product can be written as the sum of entry-wise products of elements. Tr(a(cid:62)b) = tr(ab(cid:62)) = tr(b(cid:62)a) = tr(ba(cid:62)) n(cid:88) i=1 n(cid:88)
