Multiplying and Factoring Matrices The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Strang, Gilbert. “Multiplying and Factoring Matrices.” The American mathematical monthly, vol. 125, no. 3, 2020, pp. 223-230 © 2020 The Author As Published 10.1080/00029890.2018.1408378 Publisher Informa UK Limited Version Author's final manuscript Citable link https://hdl.handle.net/1721.1/126212 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ Multiplying and Factoring Matrices Gilbert Strang MIT
[email protected] I believe that the right way to understand matrix multiplication is columns times rows : T b1 . T T AB = a1 ... an . = a1b1 + ··· + anbn . (1) bT n T Each column ak of an m by n matrix multiplies a row of an n by p matrix. The product akbk is an m by p matrix of rank one. The sum of those rank one matrices is AB. T T All columns of akbk are multiples of ak, all rows are multiples of bk . The i, j entry of this rank one matrix is aikbkj . The sum over k produces the i, j entry of AB in “the old way.” Computing with numbers, I still find AB by rows times columns (inner products)! The central ideas of matrix analysis are perfectly expressed as matrix factorizations: A = LU A = QR S = QΛQT A = XΛY T A = UΣV T The last three, with eigenvalues in Λ and singular values in Σ, are often seen as column-row multiplications (a sum of outer products).