text: # Linear Algebra with Applications (3rd Edition)
  
  • Homework Help
  • davidvictor
  • 9
  • 100% (2) 2 out of 2 people found this document helpful
  
  This preview shows page 8 - 9 out of 9 pages.
  
  ##### We have textbook solutions for you!
  The document you are viewing contains questions related to this textbook.
  The document you are viewing contains questions related to this textbook.
  Chapter 4 / Exercise 63
  Algebra and Trigonometry: Real Mathematics, Real People
  Larson
  Expert Verified
  It follows that A 2 = - 4 3 / 4 - 3 / 4 - 1 / 4 1 / 4 = - 4 P 2 is the required decomposition, where P 2 is a projection matrix.
  ##### We have textbook solutions for you!
  The document you are viewing contains questions related to this textbook.
  The document you are viewing contains questions related to this textbook.
  Chapter 4 / Exercise 63
  Algebra and Trigonometry: Real Mathematics, Real People
  Larson
  Expert Verified
  8. [10 marks] 8a. [3 marks] Find the lengths and the inner product x · y of the following complex vectors x = 2 - 4 i 4 i , y = 2 4 ( i 2 = - 1) . 8b. [3 marks] Let A = 1 1 - i 1 + i 2 . Let x 1 , x 2 be two (linearly independent) eigen- vectors of A . Compute x 1 · x 2 and show that det( A ) R . 8c. [4 marks] Prove that for any complex vector x x H Ax R . ( H = Hermitian) Sol. [8a.] length( x ) = ( x H x ) 1 / 2 = ( £ 2 + 4 i - 4 i / · 2 - 4 i 4 i ) 1 / 2 = 6; length( y ) = ( y T y ) 1 / 2 = 20 and x · y := x H y = 4(1 - 2 i ). [8b.] Notice that A = A H , furthermore let λ i be the 2 eigenvalues of A : 0 = λ 1 6 = λ 2 = 3, then x 1 · x 2 = 0. Also, one knows that every eigenvalue of a Hermitian matrix is real and so will be its determinant (det( A ) = λ 1 λ 2 = 0). [8c.] We have ( x H Ax ) H = x H Ax , as A = A H . It follows that x H Ax R .
explanation: The document is a snippet from a homework help resource for a Linear Algebra course. It contains multiple math questions and their corresponding answers, specifically dealing with complex vectors, eigenvalues, and Hermitian matrices.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Linear Algebra with Applications (3rd Edition)
  
  • Homework Help
  • davidvictor
  • 9
  • 100% (2) 2 out of 2 people found this document helpful
  
  This preview shows page 8 - 9 out of 9 pages.
  
  ##### We have textbook solutions for you!
  The document you are viewing contains questions related to this textbook.
  The document you are viewing contains questions related to this textbook.
  Chapter 4 / Exercise 63
  Algebra and Trigonometry: Real Mathematics, Real People
  Larson
  Expert Verified
  It follows that A 2 = - 4 3 / 4 - 3 / 4 - 1 / 4 1 / 4 = - 4 P 2 is the required decomposition, where P 2 is a projection matrix.
  ##### We have textbook solutions for you!
  The document you are viewing contains questions related to this textbook.
  The document you are viewing contains questions related to this textbook.
  Chapter 4 / Exercise 63
  Algebra and Trigonometry: Real Mathematics, Real People
  Larson
  Expert Verified
  8. [10 marks] 8a. [3 marks] Find the lengths and the inner product x · y of the following complex vectors x = 2 - 4 i 4 i , y = 2 4 ( i 2 = - 1) . 8b. [3 marks] Let A = 1 1 - i 1 + i 2 . Let x 1 , x 2 be two (linearly independent) eigen- vectors of A . Compute x 1 · x 2 and show that det( A ) R . 8c. [4 marks] Prove that for any complex vector x x H Ax R . ( H = Hermitian) Sol. [8a.] length( x ) = ( x H x ) 1 / 2 = ( £ 2 + 4 i - 4 i / · 2 - 4 i 4 i ) 1 / 2 = 6; length( y ) = ( y T y ) 1 / 2 = 20 and x · y := x H y = 4(1 - 2 i ). [8b.] Notice that A = A H , furthermore let λ i be the 2 eigenvalues of A : 0 = λ 1 6 = λ 2 = 3, then x 1 · x 2 = 0. Also, one knows that every eigenvalue of a Hermitian matrix is real and so will be its determinant (det( A ) = λ 1 λ 2 = 0). [8c.] We have ( x H Ax ) H = x H Ax , as A = A H . It follows that x H Ax R .
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a snippet from a homework help resource for a Linear Algebra course. It contains multiple math questions and their corresponding answers, specifically dealing with complex vectors, eigenvalues, and Hermitian matrices.
  
  Label: multiple-question-and-answer