Own favorite way to understand matrix multiplication: For any positive integer n, and any n x n matrix A of real and/or complex numbers, there are n x n matric

Own favorite way to understand matrix multiplication: For any positive integer n, and any n x n matrix A of real and/or complex numbers, there are n x n matrices U and H so that A = UH Each of U and H consists of real and/or complex numbers and all real numbers if A consists of all real numbers. Here U is unitary which means that for any n x 1 vector (real and/or complex) x, Ux is the same as x except is rotated and/or reflected and the lengths |x| = |Ux| that is, U does not change lengths or distances and, thus, is a rigid motion (rotation, reflection). For H, for x in a sphere S, the set of all Hx is just an ellipsoid. So, A = UH where U is a rigid motion, rotation, reflection, and H converts a sphere to an ellipsoid. The H is said to be Hermitian (there was a mathematician Hermite). Right, an ellipsoid has mutually perpendicular axes, and those are the eigenvectors of H. My favorite result in linear algebra, and with a fairly short and simple proof.