Prove that the product of two n x n anti-diagonal matrices is diagonal.
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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![2. A diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the upper
left corner to the lower right corner. Formally, an n x n matrix A = [aj] is diagonal if a;j = 0 whenever i + j.
Equivalently, if azj 70 then i = j.
An anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the
lower left corner to the upper right corner. Formally, an n x n matrix A = [a;] is anti-diagonal if a;j = 0 whenever
i+j+n+1. Equivalently, if azj +0 then i +j=n+1.
Prove that the product of two n x n anti-diagonal matrices is diagonal. (See outline below, also the proof in #13 of
homework 2.1)
Outline:
(1) Begin your proof by letting A = [a;j] and B = [bj] be two anti-diagonal n Xn matrices.To prove that the matrix AB is
diagonal you need to show that the ij-th entry of AB is 0 for all i + j, or, equivalently, if the ij-th entry of AB is not 0,
then i = j. (this second part is the one to use).
(ii) The ij-th entry of the product AB is cij = (write in terms of E)
(ii) Show that if Cij 70 (*) then i = j.
* HINT: there must be at least one term a¡kbrj that is not 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42f67b29-db3a-470e-8f57-9956af53c593%2F8f5cfcdb-3562-4ffc-83d1-023b78987835%2Felznglr_processed.png&w=3840&q=75)
Transcribed Image Text:2. A diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the upper
left corner to the lower right corner. Formally, an n x n matrix A = [aj] is diagonal if a;j = 0 whenever i + j.
Equivalently, if azj 70 then i = j.
An anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the
lower left corner to the upper right corner. Formally, an n x n matrix A = [a;] is anti-diagonal if a;j = 0 whenever
i+j+n+1. Equivalently, if azj +0 then i +j=n+1.
Prove that the product of two n x n anti-diagonal matrices is diagonal. (See outline below, also the proof in #13 of
homework 2.1)
Outline:
(1) Begin your proof by letting A = [a;j] and B = [bj] be two anti-diagonal n Xn matrices.To prove that the matrix AB is
diagonal you need to show that the ij-th entry of AB is 0 for all i + j, or, equivalently, if the ij-th entry of AB is not 0,
then i = j. (this second part is the one to use).
(ii) The ij-th entry of the product AB is cij = (write in terms of E)
(ii) Show that if Cij 70 (*) then i = j.
* HINT: there must be at least one term a¡kbrj that is not 0.
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