(a) Let A € Rxn be nonsingular, and assume that we know the LU decomposition ALU. Let BE Rnxk, CERnxk, and DE Rkxk. We want to compute the solution of A B D - B as efficiently as possible using the fact that we know the LU decomposition of A. (i) Give explicit expressions for a and y in terms of the matrices A, B, C, and D (and their inverses where appropriate), and the right-hand side vectors by and b₂. (ii) Show that the complexity of computing e and y is O(kn²+k²n+k³) if the LU decomposition of A is known. (b) Now let A € Rmxn, m≥n. Let trace(A) := the sum of its diagonal values). Note that || A|| -1 ajj be the trace of A (that is := [trace(ATA)]¹/2 (i) Show that trace(AB) = trace(BA) for square matrices with the same di- mensions. (ii) Conclude that ||PAQ||||A|| for orthogonal matrices PE Rmxm and QE Rnxn,

(a) Let A € Rxn be nonsingular, and assume that we know the LU decomposition ALU. Let BE Rnxk, CERnxk, and DE Rkxk. We want to compute the solution of A B D - B as efficiently as possible using the fact that we know the LU decomposition of A. (i) Give explicit expressions for a and y in terms of the matrices A, B, C, and D (and their inverses where appropriate), and the right-hand side vectors by and b₂. (ii) Show that the complexity of computing e and y is O(kn²+k²n+k³) if the LU decomposition of A is known. (b) Now let A € Rmxn, m≥n. Let trace(A) := the sum of its diagonal values). Note that || A|| -1 ajj be the trace of A (that is := [trace(ATA)]¹/2 (i) Show that trace(AB) = trace(BA) for square matrices with the same di- mensions. (ii) Conclude that ||PAQ||||A|| for orthogonal matrices PE Rmxm and QE Rnxn,

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.7: The Inverse Of A Matrix

Problem 32E

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