Chapter_09

Chapter_09 - CHAPTER 9 9.1 Assume that ( n, i ) = ( i ) (...

This preview shows pages 1–4. Sign up to view the full content.

197 CHAPTER 9 9.1 Assume that Hence, for i = n : Since β ( n,n ) = 1, we have Next, for i = n -1, or equivalently, Proceeding in this way, we may thus write For β ( n,i ) to equal λ n-i , we must have This requirement is satisfied by choosing for all k β ni , () λ i ()β -1 , , = i 1 n ,, = β nn , ()λ n β -1 , = λ 1 n β -1 , = β -1 , λ n -1 β -2 , = β -2 , λ 1 n -1 β -1 , = λ 1 n -1 λ 1 n = β , λ 1 i +1 λ 1 n -1 λ 1 n = λ 1 k k = i +1 n = λ 1 i +1 λ 1 n -1 λ 1 n λ n - i = λ k λ 1 =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
198 We thus find that 9.2 The matrix inversion lemma states that if A = B -1 + CD -1 C (1) then A -1 = B - BC ( D + C H BC ) -1 C H B (2) To prove this lemma, we multiply Eq. (1) by (2): AA -1 = ( B -1 + CD -1 C H )[ B - BC ( D + C H BC ) -1 C H B ] = B -1 B - B -1 BC ( D + C H BC ) -1 C H B - CD -1 C H BC ( D + C H BC ) -1 C H B + CD -1 C H B We have to show that AA -1 = I . Since ( D + C H BC ) -1 ( D + C H BC )= I , and B -1 B = I ,we may rewrite this result as AA -1 = I - C ( D + C H BC ) -1 C H B + CD -1 ( D + C H BC )( D + C H BC ) -1 C H B - CD -1 C H BC ( D + C H BC ) -1 C H B = I - [ C - CD -1 ( D + C H BC ) - CD -1 C H BC ] · ( D + C H BC ) -1 C H B = I - ( C - CD -1 D )( D + C H BC ) -1 C H B Since D -1 D = I, the second term in this last line is zero; hence, AA -1 = I which is the desired result. β ni , () terms λ…λλ = λ n - i =
199 9.3 We are given (1) Let A = B -1 + CD -1 C H (2) Then, according to the matrix inversion lemma: A -1 = B - BC ( D + C H BC ) -1 C H B (3) From Eqs. (1) and (2), we note: A = B -1 = δ I C = u ( n ) D = I Hence, using Eq. (3): 9.4 From Section 9.6, we have (1) Φ n () δ Iu n u H n + = Φ n Φ 1 n 1 δ -- I 1 δ 2 ----- u n 1 1 δ u H n u n +   1 u H n

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/11/2010 for the course EE EE254 taught by Professor Ujin during the Spring '10 term at YTI Career Institute.

Page1 / 10

Chapter_09 - CHAPTER 9 9.1 Assume that ( n, i ) = ( i ) (...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online