# Class 14 - ecse.rpi.edu Solving Rectangular Matrices Pseudo-inverse M. A. Hameed ECSE Department Rensselaer Polytechnic Institute Intro to ECSE Earlier in the course We solved a system of simultaneous linear equations by formulating it as: Ax=b Matrix Anxn was square and invertible Information about the circuit (system) Column vector xnx1 contained unknown quantities Column vector bnx1 contained known quantities Solve for unknowns using x = A-1b M. Hameed Friday, January 31, 2020

Earlier in the course2 Ax=b matrix view A M. Hameed x = b Friday, January 31, 2020 What is A is not a square matrix Consider A as a rectangular matrix m rows and n columns and m > n Now Ax=b matrix view is A

mxn X = b nx1 mx1 M. Hameed Friday, January 31, 2020 How to invert a rectangular matrix A We need its pseudo-inverse = = Transpose (next slide) =

1 =( ) M. Hameed Friday, January 31, 2020 Transpose of a matrix [ flips a matrix over its diagonal

write the rows of A as the columns of AT, write the columns of A as the rows of AT. Example: Also show in Matlab M. Hameed ] = [

] Friday, January 31, 2020 Revisit 1

=( ) Size of A = m x n Size of transpose(A) = n x m ATA is possible (inner dimensions match) ATA results in a square matrix of size ATA is now invertible However, we still need to satisfy det(ATA) 0 M. Hameed Friday, January 31, 2020