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We say that a matrix is upper triangular if all their entries below the main diagonal are 0, and that it is strictly upper triangular if in addition all the entries on the main diagonal are equal to 1. Any invertible real n x n matrix A can be written as the product of three real n x n matrices A = ODN where N is strictly upper triangular, D is diagonal with positive entries, and O is orthogonal. This is known as the Iwasawa decomposition of the matrix A. The decomposition is unique. In other words, that if A = O'D'N', where O', D' and N' are orthogonal, diagonal with positive entries and strictly upper triangular, respectively, then O' = 0, D = D' and N' = N. (i) Find the Iwasawa decomposition of the matrix A = (i ²). 01 12 (ii) Consider the 2 x 2 matrix M = · (ad) c where a, b, c, d e C and ad- bc = 1. Thus M is an element of the Lie group SL(2, C). The Iwasawa decomposition is given by a a В 0 (d) = ( - ) (6-1² 1/²) (17) 0 81/2 where a, ß, n = C and 8 € R+. Find a, 3, 6 and n.

We say that a matrix is upper triangular if all their entries below the main diagonal are 0, and that it is strictly upper triangular if in addition all the entries on the main diagonal are equal to 1. Any invertible real n x n matrix A can be written as the product of three real n x n matrices A = ODN where N is strictly upper triangular, D is diagonal with positive entries, and O is orthogonal. This is known as the Iwasawa decomposition of the matrix A. The decomposition is unique. In other words, that if A = O'D'N', where O', D' and N' are orthogonal, diagonal with positive entries and strictly upper triangular, respectively, then O' = 0, D = D' and N' = N. (i) Find the Iwasawa decomposition of the matrix A = (i ²). 01 12 (ii) Consider the 2 x 2 matrix M = · (ad) c where a, b, c, d e C and ad- bc = 1. Thus M is an element of the Lie group SL(2, C). The Iwasawa decomposition is given by a a В 0 (d) = ( - ) (6-1² 1/²) (17) 0 81/2 where a, ß, n = C and 8 € R+. Find a, 3, 6 and n.

Advanced Engineering Mathematics
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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We say that a matrix is upper triangular if all their entries below the main diagonal are 0, and that it is strictly upper triangular if in addition all the entries on the main diagonal are equal to 1. Any invertible real nxn matrix A can be written as the product of three real n x n matrices A = ODN where N is strictly upper triangular, D is diagonal with positive entries, and O is orthogonal. This is known as the Iwasawa decomposition of the matrix A. The decomposition is unique. In other words, that if A = O'D'N', where O', D' and N' are orthogonal, diagonal with positive entries and strictly upper triangular, respectively, then O' = 0, D = D' and N' = N. (i) Find the Iwasawa decomposition of the matrix A = (i ²). = (ii) Consider the 2 x 2 matrix M = - (ad) where a, b, c, d e C and ad- bc = 1. Thus M is an element of the Lie group SL(2, C). The Iwasawa decomposition is given by a b B (d) = (-2) (6-1² ²/2) (17) (5-1/2 -Ba 61/2 0 where a, ß, ne C and 8 € R+. Find a, 3, 6 and n.
Transcribed Image Text:We say that a matrix is upper triangular if all their entries below the main diagonal are 0, and that it is strictly upper triangular if in addition all the entries on the main diagonal are equal to 1. Any invertible real nxn matrix A can be written as the product of three real n x n matrices A = ODN where N is strictly upper triangular, D is diagonal with positive entries, and O is orthogonal. This is known as the Iwasawa decomposition of the matrix A. The decomposition is unique. In other words, that if A = O'D'N', where O', D' and N' are orthogonal, diagonal with positive entries and strictly upper triangular, respectively, then O' = 0, D = D' and N' = N. (i) Find the Iwasawa decomposition of the matrix A = (i ²). = (ii) Consider the 2 x 2 matrix M = - (ad) where a, b, c, d e C and ad- bc = 1. Thus M is an element of the Lie group SL(2, C). The Iwasawa decomposition is given by a b B (d) = (-2) (6-1² ²/2) (17) (5-1/2 -Ba 61/2 0 where a, ß, ne C and 8 € R+. Find a, 3, 6 and n.
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