(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,
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please answer the whole 2 question thank you l'll give you a thumb
![2.
(a) Let A € Rnxn be nonsingular, and assume that we know the LU decomposition
A = LU. Let B E Rnxk, CE Rnxk, and D € Rkxk
We want to compute the solution of
A
[BD-R]
as efficiently as possible using the fact that we know the LU decomposition of
A.
(i) Give explicit expressions for r 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 a 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) :=
-1 aj be the trace of A (that is
=1
the sum of its diagonal values). Note that || A|| F = [trace(ATA)]¹/2
:=
(i) Show that trace(AB) = trace (BA) for square matrices with the same di-
mensions.
(ii) Conclude that ||PAQ||F = ||A||F for orthogonal matrices P € Rmxm and
QERnxn](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2798f8b3-0049-47d1-b102-d46826ed0106%2F08c40329-95ba-4d81-a6e6-d533d5536452%2F5vht44z_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2.
(a) Let A € Rnxn be nonsingular, and assume that we know the LU decomposition
A = LU. Let B E Rnxk, CE Rnxk, and D € Rkxk
We want to compute the solution of
A
[BD-R]
as efficiently as possible using the fact that we know the LU decomposition of
A.
(i) Give explicit expressions for r 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 a 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) :=
-1 aj be the trace of A (that is
=1
the sum of its diagonal values). Note that || A|| F = [trace(ATA)]¹/2
:=
(i) Show that trace(AB) = trace (BA) for square matrices with the same di-
mensions.
(ii) Conclude that ||PAQ||F = ||A||F for orthogonal matrices P € Rmxm and
QERnxn
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